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100% equals 1 because it represents the fraction $100/100$. When the numerator and denominator of a fraction are the same nonzero number, the value of the fraction is exactly one whole.

Conditions: The percentage value is 100.; The denominator is 100.

100% equals 1 because it represents the fraction $100/100$. When the numerator and denominator of a fraction are the same nonzero number, the value of the fraction is exactly one whole.

Conditions: The percentage value is 100.; The denominator is 100.

The expected number of heads is 50 because the expected value for a counting experiment like this is calculated as the number of trials multiplied by the probability of success on each trial. With 100 tosses and a fair coin (probability of heads = 0.5), the calculation is $100 \cdot 0.5 = 50$.

Conditions: The coin is fair.; There are 100 tosses per trial.; $X$ counts heads.

The expected number of heads is 50 because the expected value for a counting experiment like this is calculated as the number of trials multiplied by the probability of success on each trial. With 100 tosses and a fair coin (probability of heads = 0.5), the calculation is $100 \cdot 0.5 = 50$.

Conditions: The coin is fair.; There are 100 tosses per trial.; $X$ counts heads.

An equality constraint is used because the customer order requires exactly 1000 units to be supplied. The problem specifies that the order must be met precisely, meaning neither underproduction (shortfall) nor overproduction (surplus) is permitted.

Conditions: The problem involves fulfilling a specific customer order.; The order quantity is fixed and must be met exactly.; Overproduction or underproduction is not allowed or desired.

An equality constraint is used because the customer order requires exactly 1000 units to be supplied. The problem specifies that the order must be met precisely, meaning neither underproduction (shortfall) nor overproduction (surplus) is permitted.

Conditions: The problem involves fulfilling a specific customer order.; The order quantity is fixed and must be met exactly.; Overproduction or underproduction is not allowed or desired.

The example concludes this because the conditional expectation of the discounted time-3 stock price, given information at time 2, equals the discounted time-2 stock price. Specifically, $\mathbb{E}_2\left[\frac{S_3}{(1+r)^3}\right] = 82.64$ and $\frac{S_2}{(1+r)^2} = 82.64$, satisfying the martingale condition $M_n = \mathbb{E}_n[M_{n+1}]$.

Conditions: The expectation is taken under the risk-neutral probability $p = 2/3$.; Discounting uses the interest rate $r = 0.1$.

The example concludes this because the conditional expectation of the discounted time-3 stock price, given information at time 2, equals the discounted time-2 stock price. Specifically, $\mathbb{E}_2\left[\frac{S_3}{(1+r)^3}\right] = 82.64$ and $\frac{S_2}{(1+r)^2} = 82.64$, satisfying the martingale condition $M_n = \mathbb{E}_n[M_{n+1}]$.

Conditions: The expectation is taken under the risk-neutral probability $p = 2/3$.; Discounting uses the interest rate $r = 0.1$.

The sample mean converges to 50 because 50 is the expected value $E(X)$ of the random variable $X$ (number of heads in 100 fair-coin tosses). According to the law of large numbers, as the number of trials $n$ approaches infinity, the sample mean $\overline{X}_n$ approaches the expected value.

Conditions: The coin is fair.; $X$ counts heads after 100 tosses.; $n$ approaches infinity.

The sample mean converges to 50 because 50 is the expected value $E(X)$ of the random variable $X$ (number of heads in 100 fair-coin tosses). According to the law of large numbers, as the number of trials $n$ approaches infinity, the sample mean $\overline{X}_n$ approaches the expected value.

Conditions: The coin is fair.; $X$ counts heads after 100 tosses.; $n$ approaches infinity.

The state prices $\psi_j$ represent the prices of Arrow-Debreu securities, which are monetary values and do not necessarily sum to 1. The transformed vector $q_j$ represents risk-neutral probabilities, obtained by multiplying each state price by the gross risk-free factor $(1+i)$.

Conditions: The market is a one-period finite-state model.; $\psi_j$ are the solved state prices (Arrow-Debreu security prices).; $i$ is the per-period risk-free interest rate.; $q_j$ is defined as $q_j = (1+i)\psi_j$.

The state prices $\psi_j$ represent the prices of Arrow-Debreu securities, which are monetary values and do not necessarily sum to 1. The transformed vector $q_j$ represents risk-neutral probabilities, obtained by multiplying each state price by the gross risk-free factor $(1+i)$.

Conditions: The market is a one-period finite-state model.; $\psi_j$ are the solved state prices (Arrow-Debreu security prices).; $i$ is the per-period risk-free interest rate.; $q_j$ is defined as $q_j = (1+i)\psi_j$.