Probability is not Likelihood

1. Formula
   Observation

   The formula $P(x \mid \theta)$ is displayed on screen.

2. Audio
   Observation

   The narrator reads it as "P of x given theta".

P

Probability notation; the conditional probability operator.

Applied to the pair $(x, \theta)$ in the video.

1. Formula
   Observation

   $x$ appears inside $P(x \mid \theta)$.

2. Animation
   Observation

   At 00:06-00:13, $x$ is boxed and labeled "vary" while $\theta$ is labeled "fix".

3. Animation
   Observation

   At 00:14-00:26, $x$ is underlined and labeled "fix" while $\theta$ is boxed and labeled "vary".

4. Audio
   Observation

   The narrator says "let x vary" for probability and "if you fix x" for likelihood.

x

Data or observed outcome variable; in the concrete example it corresponds to the number of heads.

1. Formula
   Observation

   $\theta$ appears inside $P(x \mid \theta)$.

2. Animation
   Observation

   At 00:06-00:13, $\theta$ is underlined and labeled "fix".

3. Animation
   Observation

   At 00:14-00:26, $\theta$ is boxed and labeled "vary".

4. Audio
   Observation

   The narrator says "If you fix theta" for probability and "let theta vary" for likelihood.

$\theta$

Model parameter; in the coin example it is instantiated as $p$.

1. Formula
   Observation

   $p = 0.5$ is shown at 00:29-00:33.

2. Audio
   Observation

   The narrator says "a coin with probability p equals 0.5 of landing heads".

3. Formula
   Observation

   Later lines show $p = 0.3$, $p = 0.5$, and $p = 0.7$.

4. Diagram
   Observation

   The likelihood plot uses $p$ on the horizontal axis from 0 to 1.

p

Coin-head probability parameter; the concrete value of $\theta$ used in the example.

The video plots it over values between 0 and 1.

1. Diagram
   Observation

   The vertical axis is labeled $P(k)$.

2. Formula
   Observation

   The normalization equation is $\sum_{k=0}^{10} P(k) = 1$.

3. Formula
   Observation

   A highlighted value is written as $P(7) = 0.117$.

k

Number of heads in 10 flips; the discrete outcome indexed by the probability bars.

Visible summation range is $k=0,1,\dots,10$.

1. Diagram
   Observation

   The vertical axis of the second graph is labeled $L(p)$.

2. Audio
   Observation

   The narrator calls the plotted object "the likelihood function".

$L(p)$

Likelihood function of the parameter $p$ given the fixed observation of 7 heads in 10 flips.

1. Formula
   Observation

   $\sum_{k=0}^{10} P(k) = 1$ appears on screen.

2. Audio
   Observation

   The narrator says "All these bars add up to exactly one."

$\sum_{k=0}^{10} P(k) = 1$

Normalization condition for the probability distribution over all possible head counts.

1. Diagram
   Observation

   The vertical axis of the first graph is labeled $L(p)$.

2. Formula
   Observation

   The displayed likelihood formula is $L(p) = \binom{10}{7} \cdot p^7 \cdot (1-p)^3$.

3. Audio
   Observation

   The narrator refers to the likelihood function and says it does not tell you the probability that p has a certain value.

$L(p)$

Likelihood function for the coin parameter p, evaluated at the observed data $k=7$ heads out of $n=10$ tosses.

$p \in [0,1]$

1. Diagram
   Observation

   The horizontal axis of the likelihood graph is labeled p with tick marks from 0.2 to 1.0.

2. Audio
   Observation

   The narrator says the peak sits right at $p = 0.7$ and later contrasts fixing p versus varying p.

3. Formula
   Observation

   p appears in $L(p) = \binom{10}{7} \cdot p^7 \cdot (1-p)^3$ and in the comparison text 'Fix $p = 0.5$, vary k' and 'Fix $k = 7$, vary p'.

p

Parameter representing the probability of heads for the coin; in the likelihood view it is treated as the variable being compared across values.

$0 \le p \le 1$

1. Formula
   Observation

   The displayed inequality is $\int_0^1 L(p)$\,dp $\neq 1$.

2. Audio
   Observation

   The narrator says, 'if you compute the area under this curve, it does not equal one.'

$\int_0^1 L(p)$\,dp

Integral of the likelihood function over the full parameter range $p \in [0,1]$; used to show that likelihood is not normalized like a probability density.

1. Formula
   Observation

   The binomial coefficient $\binom{10}{7}$ appears in the displayed likelihood formula.

2. Audio
   Observation

   The narrator says, 'For our coin, L of p equals the binomial coefficient times p to the seven times one minus p to the three.'

$\binom{10}{7}$

Binomial coefficient counting the number of ways to obtain 7 heads in 10 tosses.

1. Formula
   Observation

   The factor $p^7$ appears in $L(p) = \binom{10}{7} \cdot p^7 \cdot (1-p)^3$.

2. Audio
   Observation

   The narrator reads the formula aloud as 'times p to the seven'.

$p^7$

Probability contribution from observing 7 heads when each head occurs with probability p.

1. Formula
   Observation

   $P(x \mid \theta)$ is shown throughout the opening segment.

2. Animation
   Observation

   The labels "fix" and "vary" switch between $\theta$ and $x$.

3. Audio
   Observation

   The narrator states that the same formula means two different things depending on what is fixed.

The video defines the distinction operationally: if $\theta$ is fixed and $x$ varies, $P(x \mid \theta)$ is treated as a probability over data; if $x$ is fixed and $\theta$ varies, the same expression is treated as a likelihood over model parameters.

1. For probability: $\theta$ is fixed and $x$ varies.

2. For likelihood: $x$ is fixed and $\theta$ varies.

1. Animation
   Observation

   $x$ is boxed with "vary" and $\theta$ is underlined with "fix".

2. Caption evidence
   Observation

   The word "Probability" appears below the formula.

3. Audio
   Observation

   The narrator asks, "How likely is this data, given a known model?"

When the model parameter is fixed, the expression measures how likely each possible data outcome is under that known model.

1. Parameter/model is fixed.

2. Data/outcome is allowed to vary.

1. Same conditional expression can denote probability or likelihood

1. Animation
   Observation

   $x$ is underlined with "fix" and $\theta$ is boxed with "vary".

2. Caption evidence
   Observation

   The word "Likelihood" appears below the formula.

3. Audio
   Observation

   The narrator asks, "How plausible is this model, given the data we already observed?"

When the data are fixed, the expression measures how plausible different parameter values are as explanations of the observed data.

1. Observed data are fixed.

2. Parameter/model is allowed to vary.

1. Same conditional expression can denote probability or likelihood

1. Diagram
   Observation

   A bar chart with horizontal axis "heads" from 1 to 10 and vertical axis $P(k)$ is drawn.

2. Audio
   Observation

   The narrator says each bar is the probability of that specific number of heads.

3. Formula
   Observation

   $p = 0.5$ remains displayed above the chart.

1. The visible tick labels start at 1, but the summation and narration include 0 heads.

With $p=0.5$ fixed, the video represents the probabilities of different numbers of heads in 10 flips as separate bars indexed by the outcome count.

1. The coin model is fixed at $p=0.5$.

2. The number of flips is 10.

3. $k$ indexes possible head counts.

1. Probability as plausibility of data under a known model

1. Formula
   Observation

   $\sum_{k=0}^{10} P(k) = 1$ is shown on screen.

2. Audio
   Observation

   The narrator says all bars add up to exactly one and that this makes it a probability distribution.

The video identifies the defining property of the displayed probability distribution: the probabilities of all possible outcomes sum to 1.

1. The outcomes $k=0,1,\dots,10$ exhaust the sample space for the number of heads in 10 flips.

1. Probability distribution over head counts for 10 fair-coin flips

1. Caption evidence
   Observation

   The title "The Likelihood Curve" appears.

2. Diagram
   Observation

   A graph with horizontal axis $p$ and vertical axis $L(p)$ is drawn.

3. Audio
   Observation

   The narrator says, "This is the likelihood function."

After fixing the observation at 7 heads in 10 flips, the video plots $L(p)$ against $p$ so that the height at each $p$ shows how plausible that parameter value is given the data.

1. The data are fixed at 7 heads in 10 flips.

2. $p$ ranges over values between 0 and 1.

1. Likelihood as plausibility of a model given observed data

1. Diagram
   Observation

   A graph titled 'The Likelihood Curve' shows an orange bell-shaped curve on axes labeled $L(p)$ vertically and p horizontally.

2. Audio
   Observation

   The narrator says the peak sits at $p = 0.7$ and explains that the likelihood tells how well each value of p explains the data.

3. Formula
   Observation

   The displayed inequality $\int_0^1 L(p)$\,dp $\neq 1$ appears beside the shaded area under the curve.

The video presents $L(p)$ as a curve over the parameter $p \in [0,1]$. Its height measures how strongly each candidate value of p supports the observed data, with the largest value at $p=0.7$ in the example shown.

1. The parameter p is restricted to [0,1].

2. The curve is evaluated for fixed observed data.

1. Formula
   Observation

   The screen displays $\int_0^1 L(p)$\,dp $\neq 1$.

2. Caption evidence
   Observation

   The text 'Not a probability distribution!' appears below the graph.

3. Audio
   Observation

   The narrator states, 'The likelihood function is not a probability distribution. It doesn't tell you the probability that p has a certain value.'

A likelihood function assigns relative support to different parameter values after the data are fixed. Unlike a probability distribution over outcomes, it is not normalized so that its total area equals 1.

1. Applies to $L(p)$ viewed as a function of the parameter p with data held fixed.

1. Likelihood curve as a function of the parameter

1. Caption evidence
   Observation

   A title card reads 'Maximum Likelihood'.

2. Audio
   Observation

   The narrator says, 'the natural thing to do is pick the one at the top. That's the maximum likelihood estimate.'

3. Formula
   Observation

   The displayed result is $\hat{p}_{\text{MLE}} = \frac{7}{10} = 0.7$.

The maximum likelihood estimate is the parameter value that gives the largest likelihood for the observed data. In the coin example, the chosen estimate is the peak of the likelihood curve.

1. There must be a likelihood function over candidate parameter values.

2. The estimate is selected after the data are fixed.

1. Likelihood curve as a function of the parameter

1. Formula
   Observation

   The screen displays $L(p) = \binom{10}{7} \cdot p^7 \cdot (1-p)^3$.

2. Audio
   Observation

   The narrator says, 'For our coin, L of p equals the binomial coefficient times p to the seven times one minus p to the three.'

3. Diagram
   Observation

   The graph below the formula shows the likelihood rising, peaking near $p=0.7$, and falling again.

For the coin example, the likelihood of observing 7 heads in 10 tosses is written as a binomial expression in p. The factor $\binom{10}{7}$ counts arrangements, $p^7$ accounts for the heads, and (1-p)^3 accounts for the tails.

1. $n=10$ independent tosses are assumed.

2. The observed count is $k=7$ heads.

3. p is the probability of heads on a single toss.

1. Likelihood curve as a function of the parameter
2. $\binom{10}{7}$
3. $p^7$
4. (1-p)^3

1. Caption evidence
   Observation

   The left panel is labeled 'Probability' with 'Fix $p = 0.5$, vary k'; the right panel is labeled 'Likelihood' with 'Fix $k = 7$, vary p'.

2. Audio
   Observation

   The narrator says the only difference is which variable you hold still and which one you move.

3. Formula
   Observation

   The shared expression $P(x \mid \theta)$ is displayed between the two panels.

The video contrasts probability and likelihood by showing that both can be read from the same conditional expression $P(x \mid \theta)$. In the probability view, the parameter is fixed and the data vary; in the likelihood view, the data are fixed and the parameter varies.

1. One view fixes the parameter and varies the data.

2. The other view fixes the data and varies the parameter.

1. Likelihood curve as a function of the parameter
2. Maximum likelihood estimate
3. $P(x \mid \theta)$

1. Audio
   Observation

   The narrator says, "Here's a formula that means two completely different things."

2. Caption evidence
   Observation

   The text "Same formula. Different meaning." appears.

The expression $P(x \mid \theta)$ can mean probability when $\theta$ is fixed and $x$ varies, and likelihood when $x$ is fixed and $\theta$ varies.

1. The same symbolic expression $P(x \mid \theta)$ is being considered.

2. One of $x$ or $\theta$ is designated fixed while the other varies.

For the displayed formula, changing which argument is fixed changes the interpretation.

1. Formula
   Observation

   $\sum_{k=0}^{10} P(k) = 1$ is shown.

2. Audio
   Observation

   The narrator says, "All these bars add up to exactly one. That's what makes it a probability distribution."

For the displayed probability model, the probabilities of all possible head counts sum to 1.

1. The bars represent $P(k)$ for all possible outcomes $k=0,\dots,10$ in 10 flips.

Over all $k$ from 0 to 10.

1. Audio
   Observation

   The narrator says, "For each value of p on the horizontal axis, the height tells you how plausible that p is, given that we saw seven heads out of ten."

2. Diagram
   Observation

   The graph is labeled $L(p)$ versus $p$.

On the likelihood plot, the height of the curve at a given $p$ measures how plausible that $p$ is given the observed data of 7 heads in 10 flips.

1. The observation is fixed at 7 heads in 10 flips.

2. The horizontal axis represents candidate values of $p$.

For each plotted value of $p$ on the horizontal axis.

1. Formula
   Observation

   The displayed inequality is $\int_0^1 L(p)$\,dp $\neq 1$.

2. Audio
   Observation

   The narrator says, 'if you compute the area under this curve, it does not equal one.'

3. Caption evidence
   Observation

   The text 'Not a probability distribution!' appears on screen.

For the likelihood function shown, the integral over p from 0 to 1 does not equal 1, so $L(p)$ is not a probability distribution over p.

1. $L(p)$ is the likelihood function for the coin example.

2. The parameter space is $p \in [0,1]$.

For the displayed likelihood function $L(p)$ on $0 \le p \le 1$.

1. Audio
   Observation

   The narrator says, 'It doesn't tell you the probability that p has a certain value. It tells you how well each value of p explains the data.'

The likelihood function measures how well each value of p explains the observed data, rather than giving the probability that p itself takes that value.

1. The data are fixed.

2. p is treated as the parameter being compared.

For each candidate value of p in the parameter space.

1. Audio
   Observation

   The narrator says, 'the natural thing to do is pick the one at the top. That's the maximum likelihood estimate.'

2. Formula
   Observation

   The displayed result is $\hat{p}_{\text{MLE}} = \frac{7}{10} = 0.7$.

3. Animation
   Observation

   A moving point travels along the likelihood curve and stops at the peak labeled $p = 0.70$.

The maximum likelihood estimate is obtained by choosing the parameter value at the top of the likelihood curve; in the example this value is 0.7.

1. A likelihood function $L(p)$ is available.

2. The example uses the displayed binomial likelihood.

For the shown likelihood curve.

1. Formula
   Observation

   The shared expression $P(x \mid \theta)$ is displayed between the two comparison panels.

2. Audio
   Observation

   The narrator says, 'Same formula underneath both. P of x given theta. The only difference is which variable you hold still and which one you move.'

3. Caption evidence
   Observation

   The left panel says 'Fix $p = 0.5$, vary k' and the right panel says 'Fix $k = 7$, vary p'.

Probability and likelihood are presented as two readings of the same underlying expression $P(x \mid \theta)$, distinguished by whether the data or the parameter is varied.

1. The comparison uses the same underlying conditional expression.

2. One view fixes the parameter and varies the data; the other fixes the data and varies the parameter.

For the generic expression $P(x \mid \theta)$ as used in the comparison.

1. Formula
   Observation

   The left panel displays $\sum = 1$ beneath the probability bar chart.

2. Audio
   Observation

   The narrator says, 'The bars sum to one.'

3. Caption evidence
   Observation

   The left panel text reads 'Fix $p = 0.5$, vary k'.

1. The explicit summand and index are not shown on screen in this later panel.

When the parameter is fixed and the possible outcomes are varied, the resulting probability distribution is normalized so that the probabilities sum to 1.

1. p is fixed at 0.5 in the example.

2. All possible outcomes k are included.

For the displayed probability distribution over k.

1. Formula
   Observation

   The right panel displays $\int \neq 1$ beneath the likelihood curve.

2. Audio
   Observation

   The narrator says, 'The curve does not integrate to one.'

3. Caption evidence
   Observation

   The right panel text reads 'Fix $k = 7$, vary p'.

1. The explicit limits and integrand are omitted in this later panel, though the earlier panel showed $\int_0^1 L(p)$\,dp $\neq 1$.

When the data are fixed and the parameter is varied, the likelihood curve is not normalized and does not integrate to 1 over the parameter space.

1. k is fixed at 7 in the example.

2. p is varied over its parameter space.

For the displayed likelihood function over p.

1. Formula
   Observation

   The screen lists $p = 0.3 \rightarrow 0.0090$, $p = 0.5 \rightarrow 0.1172$, and $p = 0.7 \rightarrow 0.2668$.

2. Audio
   Observation

   The narrator compares the three values and concludes that 0.7 looks like the best candidate.

3. Animation
   Observation

   The line for $p=0.7$ is boxed at the end.

1. The video does not display the binomial coefficient formula or the full algebraic computation behind the numbers.

1. Expression

   $$p = 0.3 \rightarrow 0.0090$$
   Explanation

   The video evaluates the fixed observation under the candidate $p=0.3$ and reports 0.0090.

   Justification

   Directly shown on screen and spoken by the narrator.

   Shown in the video
2. Expression

   $$p = 0.5 \rightarrow 0.1172$$
   Explanation

   The same observation is evaluated under $p=0.5$, giving a larger value, 0.1172.

   Justification

   Directly shown on screen and spoken by the narrator.

   Shown in the video
3. Expression

   $$p = 0.7 \rightarrow 0.2668$$
   Explanation

   The observation is evaluated under $p=0.7$, giving the largest of the three listed values.

   Justification

   Directly shown on screen and spoken by the narrator.

   Shown in the video
4. Expression

   $$0.7 \text{ is the best candidate among } \{0.3,0.5,0.7\}$$
   Explanation

   Because 0.2668 is larger than 0.1172 and 0.0090, the video identifies $p=0.7$ as the best candidate from the sampled values.

   Justification

   Comparison of the three displayed numerical outputs.

   Shown in the video

Among the three tested values, $p=0.7$ gives the highest likelihood for observing 7 heads in 10 flips.

1. Audio
   Observation

   The narrator says, "Instead of checking just a handful of values, let's plot this for every possible p between 0 and 1."

2. Diagram
   Observation

   The scene changes from three listed values to a continuous graph titled "The Likelihood Curve".

1. The exact functional form used to generate the curve is not written on screen.

1. Expression

   $$\text{sample } p=0.3,0.5,0.7$$
   Explanation

   The video first checks only a few candidate parameter values.

   Justification

   Shown in the preceding comparison list.

   Shown in the video
2. Expression

   $$\text{plot } L(p) \text{ for } 0 \le p \le 1$$
   Explanation

   It then replaces the finite list with a continuous plot over the whole parameter interval.

   Justification

   Narrator explicitly announces plotting for every possible $p$ between 0 and 1.

   Shown in the video
3. Expression

   $$\text{peak near } p=0.7$$
   Explanation

   The resulting curve rises to a maximum around $p=0.7$, matching the earlier numerical comparison.

   Justification

   Visible shape of the plotted curve and prior highlighted value.

   Shown in the video

The likelihood curve extends the discrete comparison into a continuous visualization of plausibility across all $p\in[0,1]$.

1. Animation
   Observation

   A red point slides along the orange likelihood curve from low p to high p, with labels such as $p = 0.17$, $p = 0.38$, $p = 0.59$, and finally $p = 0.70$ at the peak.

2. Formula
   Observation

   The formula $L(p) = \binom{10}{7} \cdot p^7 \cdot (1-p)^3$ remains displayed above the graph.

3. Audio
   Observation

   The narrator says the likelihood rises, peaks at 0.7, and falls back down, then identifies the MLE as $7/10 = 0.7$.

1. Expression

   $$$$
   Explanation

   Start with the displayed likelihood function $L(p)$ for the coin example.

   Justification

   The formula is shown on screen before the animation begins.

   Shown in the video
2. Expression

   $$$$
   Explanation

   Move p from left to right along the horizontal axis and watch the corresponding height of the curve.

   Justification

   The animation traces a point along the curve as p changes.

   Shown in the video
3. Expression

   $$$$
   Explanation

   Observe that the curve first rises, reaches a highest point near $p=0.7$, and then falls.

   Justification

   The narrator explicitly describes this rise-peak-fall pattern while the moving point demonstrates it.

   Shown in the video
4. Expression

   $$\hat{p}_{\text{MLE}} = \frac{7}{10} = 0.7$$
   Explanation

   Select the parameter value at the top of the curve as the maximum likelihood estimate.

   Justification

   The narrator defines the MLE as picking the one at the top, and the final displayed formula gives the value.

   Shown in the video

For the shown likelihood curve, the maximum likelihood estimate is the peak location $p=0.7$, written as $\hat{p}_{\text{MLE}} = 7/10 = 0.7$.

1. Diagram
   Observation

   Two side-by-side panels appear: a blue bar chart on the left and an orange curve on the right.

2. Caption evidence
   Observation

   Left text: 'Probability' and 'Fix $p = 0.5$, vary k'. Right text: 'Likelihood' and 'Fix $k = 7$, vary p'.

3. Formula
   Observation

   The left normalization is $\sum = 1$, the right non-normalization is $\int \neq 1$, and the shared middle expression is $P(x \mid \theta)$.

4. Audio
   Observation

   The narrator explains that the only difference is which variable is held still and which one is moved.

1. The generic symbols x and $\theta$ are introduced visually without a separate spoken definition in this clip.

1. Expression

   $$$$
   Explanation

   Place the probability view and the likelihood view side by side.

   Justification

   The video introduces a split-screen comparison titled 'Two Sides of One Coin'.

   Shown in the video
2. Expression

   $$$$
   Explanation

   In the left panel, fix the parameter $p=0.5$ and vary the outcome count k.

   Justification

   The caption explicitly reads 'Fix $p = 0.5$, vary k'.

   Shown in the video
3. Expression

   $$\sum = 1$$
   Explanation

   Because all possible outcomes are being considered, the probability masses sum to 1.

   Justification

   The narrator says the bars sum to one and the formula $\sum = 1$ is displayed.

   Shown in the video
4. Expression

   $$$$
   Explanation

   In the right panel, fix the observed data $k=7$ and vary the parameter p.

   Justification

   The caption explicitly reads 'Fix $k = 7$, vary p'.

   Shown in the video
5. Expression

   $$\int \neq 1$$
   Explanation

   Because the curve is a likelihood over parameter values rather than a probability distribution over outcomes, it does not integrate to 1.

   Justification

   The narrator says the curve does not integrate to one and the formula $\int \neq 1$ is displayed.

   Shown in the video
6. Expression

   $$P(x \mid \theta)$$
   Explanation

   Both panels are presented as different readings of the same underlying conditional expression.

   Justification

   The narrator says 'Same formula underneath both. P of x given theta.' and the expression is shown between the panels.

   Shown in the video

The clip’s comparison shows that probability and likelihood use the same basic expression $P(x \mid \theta)$, but probability varies the data with the parameter fixed and is normalized, whereas likelihood varies the parameter with the data fixed and is not normalized.

1. Audio
   Observation

   The narrator says, "Say you have a coin with probability p equals 0.5 of landing heads. You flip it 10 times."

2. Formula
   Observation

   $p = 0.5$ is displayed.

3. Diagram
   Observation

   A bar chart over head counts is shown.

4. Formula
   Observation

   $P(7) = 0.117$ is highlighted.

1. The exact binomial formula is not written out on screen.

2. The visible x-axis tick labels begin at 1 even though the narration includes 0 heads.

Given a coin with $p=0.5$ for heads, find the probabilities of different numbers of heads in 10 flips, especially exactly 7 heads.

1. $p = 0.5$

2. Number of flips = 10

3. Outcome of interest includes 0, 1, 5, and 7 heads

Compute and visualize the probability distribution over the number of heads.

1. Expression

   $$\text{Fix } p = 0.5$$
   Explanation

   The model parameter is fixed before asking about data outcomes.

   Justification

   This matches the video's definition of probability as the case where $\theta$ is fixed and $x$ varies.

   Shown in the video
2. Expression

   $$\text{Enumerate } k = 0,1,\dots,10$$
   Explanation

   The possible numbers of heads are represented as separate bars.

   Justification

   The narrator asks about 0 heads, 1 head, 5 heads, and the chart displays bars for head counts.

   Shown in the video
3. Expression

   $$\sum_{k=0}^{10} P(k) = 1$$
   Explanation

   The probabilities of all head counts sum to 1.

   Justification

   Explicit formula and narration identify this as the key property of a probability distribution.

   Shown in the video
4. Expression

   $$P(7) = 0.117$$
   Explanation

   The probability of exactly 7 heads is read off as about 0.117.

   Justification

   Highlighted on screen and stated by the narrator.

   Shown in the video

With $p=0.5$ fixed, the probability of exactly 7 heads in 10 flips is about $0.117$, and all outcome probabilities sum to 1.

The video verifies the distribution by showing the normalization equation $\sum_{k=0}^{10} P(k)=1$.

1. Caption evidence
   Observation

   "Flip the Question" and "Observed: 7 heads in 10 flips" appear.

2. Diagram
   Observation

   Seven H circles and three T circles are shown.

3. Formula
   Observation

   The screen lists $p = 0.3 \rightarrow 0.0090$, $p = 0.5 \rightarrow 0.1172$, and $p = 0.7 \rightarrow 0.2668$.

4. Diagram
   Observation

   A likelihood curve $L(p)$ over $p\in[0,1]$ is plotted.

1. The underlying likelihood formula is not explicitly written.

2. The curve peak is visually near 0.7 but the exact maximizing value is not printed.

Given the fixed observation of 7 heads in 10 flips, compare candidate values of $p$ and identify which parameter value is most plausible.

1. Observed data: 7 heads in 10 flips

2. Candidate values checked: $p=0.3,0.5,0.7$

3. Parameter range plotted: $0\le p\le 1$

Assess which $p$ best explains the observed data and visualize likelihood across all $p$.

1. Expression

   $$\text{Fix the data: } 7 \text{ heads in } 10 \text{ flips}$$
   Explanation

   The observation is treated as fixed evidence.

   Justification

   The narrator says, "That's your data. It's fixed. It happened."

   Shown in the video
2. Expression

   $$p = 0.3 \rightarrow 0.0090$$
   Explanation

   Evaluate the observation under $p=0.3$.

   Justification

   Displayed on screen and spoken by the narrator.

   Shown in the video
3. Expression

   $$p = 0.5 \rightarrow 0.1172$$
   Explanation

   Evaluate the observation under $p=0.5$.

   Justification

   Displayed on screen and spoken by the narrator.

   Shown in the video
4. Expression

   $$p = 0.7 \rightarrow 0.2668$$
   Explanation

   Evaluate the observation under $p=0.7$.

   Justification

   Displayed on screen and spoken by the narrator.

   Shown in the video
5. Expression

   $$\text{Plot } L(p) \text{ for } 0 \le p \le 1$$
   Explanation

   Replace the finite comparison with a continuous likelihood curve.

   Justification

   Narrator explicitly says to plot for every possible $p$ between 0 and 1.

   Shown in the video

Among the sampled values, $p=0.7$ is the best candidate because it gives the largest likelihood, 0.2668; the full curve shows plausibility across all $p\in[0,1]$.

The conclusion is supported by direct numerical comparison of the three displayed likelihood values and by the peak of the plotted curve near $p=0.7$.

1. Formula
   Observation

   The displayed likelihood is $L(p) = \binom{10}{7} \cdot p^7 \cdot (1-p)^3$.

2. Audio
   Observation

   The narrator says, 'For our coin...' and later, 'The maximum likelihood estimate is p hat equals seven over ten, which is zero point seven.'

3. Animation
   Observation

   A point moves along the curve and stops at the peak labeled $p = 0.70$.

Given a coin tossed 10 times with 7 heads observed, write the likelihood as a function of the head probability p and identify the maximum likelihood estimate.

1. Number of tosses $n = 10$.

2. Observed number of heads $k = 7$.

3. p denotes the probability of heads.

Form the likelihood $L(p)$ and find $\hat{p}_{\text{MLE}}$.

1. Expression

   $$L(p) = \binom{10}{7} \cdot p^7 \cdot (1-p)^3$$
   Explanation

   Write the likelihood for observing 7 heads and 3 tails in 10 tosses as a function of p.

   Justification

   This is the formula explicitly displayed and read aloud in the video.

   Shown in the video
2. Expression

   $$$$
   Explanation

   View $L(p)$ as a curve over $p \in [0,1]$ and locate its highest point.

   Justification

   The animation traces the curve and the narrator describes it rising, peaking, and falling.

   Shown in the video
3. Expression

   $$\hat{p}_{\text{MLE}} = \frac{7}{10} = 0.7$$
   Explanation

   Choose the p-value at the peak of the likelihood curve as the maximum likelihood estimate.

   Justification

   The narrator defines the MLE as picking the one at the top, and the final displayed equation gives the value.

   Shown in the video

$\hat{p}_{\text{MLE}} = 0.7$

The answer matches the peak label $p = 0.70$ shown on the graph and the displayed equation $\hat{p}_{\text{MLE}} = 7/10 = 0.7$.

1. Caption evidence
   Observation

   Title card reads 'Two Sides of One Coin'.

2. Diagram
   Observation

   Left panel shows a blue bar chart; right panel shows an orange likelihood curve.

3. Caption evidence
   Observation

   Left text: 'Probability', 'Fix $p = 0.5$, vary k'. Right text: 'Likelihood', 'Fix $k = 7$, vary p'.

4. Formula
   Observation

   The displayed relations are $\sum = 1$, $P(x \mid \theta)$, and $\int \neq 1$.

5. Audio
   Observation

   The narrator explains the contrast between fixing p and varying k versus fixing k and varying p.

1. The example uses generic x and $\theta$ in the final shared formula without further definition in this clip.

Compare the probability view and the likelihood view for the same coin model.

1. Probability view: p is fixed at 0.5 and k varies.

2. Likelihood view: k is fixed at 7 and p varies.

3. Both views are based on the same conditional expression $P(x \mid \theta)$.

Explain how the same formula yields a normalized probability distribution on one side and a non-normalized likelihood curve on the other.

1. Expression

   $$$$
   Explanation

   On the probability side, fix $p=0.5$ and examine all possible outcome counts k.

   Justification

   The left panel caption states 'Fix $p = 0.5$, vary k'.

   Shown in the video
2. Expression

   $$\sum = 1$$
   Explanation

   The probabilities of all possible outcomes add to 1.

   Justification

   The narrator says the bars sum to one and the formula $\sum = 1$ is shown.

   Shown in the video
3. Expression

   $$$$
   Explanation

   On the likelihood side, fix the observed data $k=7$ and examine all possible parameter values p.

   Justification

   The right panel caption states 'Fix $k = 7$, vary p'.

   Shown in the video
4. Expression

   $$\int \neq 1$$
   Explanation

   The likelihood curve does not integrate to 1 over p.

   Justification

   The narrator says the curve does not integrate to one and the formula $\int \neq 1$ is shown.

   Shown in the video
5. Expression

   $$P(x \mid \theta)$$
   Explanation

   Recognize that both sides are different readings of the same underlying expression.

   Justification

   The narrator says 'Same formula underneath both. P of x given theta.'

   Shown in the video

Probability fixes the parameter and varies the data, producing a normalized distribution; likelihood fixes the data and varies the parameter, producing a non-normalized curve, even though both come from $P(x \mid \theta)$.

The conclusion matches the side-by-side captions, the displayed normalization contrast $\sum = 1$ versus $\int \neq 1$, and the shared formula $P(x \mid \theta)$.

1. Animation
   Observation

   Boxes, underlines, and the words "fix" and "vary" move between $x$ and $\theta$.

2. Caption evidence
   Observation

   "Probability" appears first, then "Likelihood", then "Same formula. Different meaning."

1. Formula $P(x \mid \theta)$

2. Labels "fix" and "vary"

3. Words "Probability" and "Likelihood"

1. First $\theta$ is marked fixed and $x$ is marked varying.

2. Then $x$ is marked fixed and $\theta$ is marked varying.

3. The caption changes from Probability to Likelihood.

1. The symbolic expression itself remains $P(x \mid \theta)$.

The visual change encodes that the mathematical meaning depends on which argument is held constant.

1. Diagram
   Observation

   Axes labeled $P(k)$ and heads are drawn, then teal bars appear.

2. Formula
   Observation

   $\sum_{k=0}^{10} P(k)=1$ appears beside the chart.

3. Animation
   Observation

   The bar for 7 heads is highlighted and labeled $P(7)=0.117$.

1. The visible tick labels start at 1 although the formula sums from 0.

1. Horizontal axis "heads"

2. Vertical axis $P(k)$

3. Teal probability bars

4. Equation $\sum_{k=0}^{10} P(k)=1$

5. Highlighted bar at 7

1. Axes are drawn first.

2. Bars populate across head counts.

3. The normalization equation appears.

4. The 7-head bar is highlighted with its value.

1. $p=0.5$ stays fixed above the chart.

The chart visualizes probabilities of data outcomes under a fixed model.

1. Diagram
   Observation

   Seven brown circles labeled H and three gray circles labeled T appear in a row.

2. Caption evidence
   Observation

   Text reads "Observed: 7 heads in 10 flips" and later "Which p produced this?"

3. Formula
   Observation

   Three candidate evaluations are listed beneath.

1. Seven H circles

2. Three T circles

3. Candidate equations for $p=0.3,0.5,0.7$

1. The observed sequence is introduced.

2. Candidate parameter evaluations are added one by one.

3. The $p=0.7$ line is boxed.

1. The observed data remain 7 heads and 3 tails.

The picture emphasizes that the data are now fixed while the parameter is being compared.

1. Diagram
   Observation

   Axes labeled $p$ and $L(p)$ are drawn.

2. Animation
   Observation

   A gold curve is traced from left to right, peaking near 0.7.

1. The exact maximum point is not numerically labeled.

1. Horizontal axis $p$ from 0 to 1

2. Vertical axis $L(p)$

3. Gold curve

1. Axes appear first.

2. The curve is drawn across the parameter interval.

3. The peak forms near $p=0.7$.

1. The observation remains fixed at 7 heads in 10 flips.

The curve shows how likelihood varies continuously with the parameter rather than only at sampled values.

1. Animation
   Observation

   A red dot appears at the top of the orange curve.

2. Animation
   Observation

   A dashed vertical line drops from the peak to the horizontal axis.

3. Caption evidence
   Observation

   The label $p = 0.7$ appears above the peak.

1. Orange likelihood curve

2. Red peak dot

3. Dashed vertical guide line

4. Label $p = 0.7$

5. Axes labeled $L(p)$ and p

1. A point is added at the maximum of the curve.

2. A vertical guide line is drawn from the peak down to the p-axis.

3. The numeric label $p = 0.7$ appears.

1. The curve shape remains the same.

2. The axes remain labeled $L(p)$ and p.

The visual emphasizes that the likelihood is maximized at $p=0.7$ for the displayed data.

1. Animation
   Observation

   The region under the orange curve becomes shaded brown.

2. Formula
   Observation

   The inequality $\int_0^1 L(p)$\,dp $\neq 1$ appears to the left of the graph.

3. Caption evidence
   Observation

   The text 'Not a probability distribution!' appears below the graph.

1. Shaded region under the curve

2. Inequality $\int_0^1 L(p)$\,dp $\neq 1$

3. Caption 'Not a probability distribution!'

1. The area under the curve is filled in.

2. A mathematical inequality is displayed next to the shaded region.

3. A warning caption appears below the graph.

1. The curve itself and the axes remain unchanged.

The shading turns the verbal claim about area into a visual object, supporting the point that likelihood is not normalized like a probability distribution.

1. Animation
   Observation

   A red point moves along the orange curve from left to right.

2. Caption evidence
   Observation

   Intermediate labels such as $p = 0.17$, $p = 0.38$, $p = 0.59$ appear during the motion.

3. Caption evidence
   Observation

   The motion ends at $p = 0.70$ at the peak.

1. Some intermediate numeric labels are only briefly visible and may be approximate due to motion.

1. Moving red point

2. Orange likelihood curve

3. Changing p-labels

1. The point travels from lower p values toward higher p values.

2. The displayed p-label updates as the point moves.

3. The point stops at the peak value $p = 0.70$.

1. The underlying likelihood formula remains displayed above the graph.

2. The axes remain fixed.

The animation demonstrates how $L(p)$ changes as p varies and visually identifies the maximizing parameter value.

1. Diagram
   Observation

   A blue bar chart appears on the left and an orange curve appears on the right.

2. Caption evidence
   Observation

   Left heading 'Probability' with 'Fix $p = 0.5$, vary k'; right heading 'Likelihood' with 'Fix $k = 7$, vary p'.

3. Formula
   Observation

   The left side shows $\sum = 1$, the right side shows $\int \neq 1$, and the center shows $P(x \mid \theta)$.

4. Animation
   Observation

   Arrows later point from the central expression toward the left and right normalization statements.

1. Blue probability bar chart

2. Orange likelihood curve

3. Left caption 'Fix $p = 0.5$, vary k'

4. Right caption 'Fix $k = 7$, vary p'

5. Central formula $P(x \mid \theta)$

6. Left formula $\sum = 1$

7. Right formula $\int \neq 1$

8. Directional arrows

1. The screen splits into two panels.

2. A discrete distribution is shown on the left and a continuous-looking curve on the right.

3. Normalization statements are added beneath each panel.

4. A shared formula is inserted between them and linked by arrows.

1. The left panel consistently represents probability and the right panel consistently represents likelihood.

The visual layout encodes the main conceptual distinction: same underlying expression, different variable held fixed, and different normalization behavior.

1. Audio
   Observation

   The narrator says the formula means "two completely different things".

2. Caption evidence
   Observation

   "Same formula. Different meaning." is shown.

Seeing the same expression $P(x \mid \theta)$ and assuming it always has the same meaning.

The video stresses that the meaning changes depending on whether $x$ or $\theta$ is fixed.

1. Formula
   Observation

   Only the probability bars are explicitly said to sum to 1.

2. Audio
   Observation

   The likelihood section describes heights as plausibility, not as probabilities over outcomes.

1. The video does not explicitly state that likelihood values need not sum or integrate to 1.

Assuming the likelihood curve $L(p)$ is itself a probability distribution over $p$ because it visually resembles a distribution.

The analyst adds: The video explicitly assigns the normalization property only to the probability distribution P of k; the interpretation of L of p is plausibility of the parameter, not a stated probability distribution.

1. Audio
   Observation

   The narrator says, 'It doesn't tell you the probability that p has a certain value.'

2. Caption evidence
   Observation

   The text 'Not a probability distribution!' is displayed.

3. Formula
   Observation

   The inequality $\int_0^1 L(p)$\,dp $\neq 1$ is shown.