Why do the bars in the first graph add up to 1?

Conditions

• The coin model is fixed at $p=0.5$.
• The number of flips is 10.
• $k$ indexes all possible head counts from 0 to 10.

Reasoning, step by step

1. Identify that the bar chart displays $P(k)$ for each possible number of heads $k$.
2. Recognize that the outcomes $k=0, 1, \dots, 10$ cover all possibilities for 10 flips.
3. Apply the normalization condition for discrete probability distributions.
4. Conclude that the sum of all probabilities must be exactly 1.

Example

The video displays the equation $\sum_{k=0}^{10} P(k) = 1$ next to the bar chart, and the narrator states, "All these bars add up to exactly one. That's what makes it a probability distribution."

Common misconceptions

• Assuming that a visual bar chart automatically represents a normalized probability distribution without checking the sum.
• Confusing the sum of probabilities with the sum of frequencies or raw counts.

Watch the explanation

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