AP Statistics 4.2 Estimating Probabilities Using Simulation- FRQs - Exam Style Questions
▶️ Answer/Explanation
(a)
(i) Let \( X \) = the number of gift cards a particular employee receives in a \( 52 \)-week year.
\( X \) follows a binomial distribution with \( n = 52 \) trials and probability of success \( p = \frac{1}{200} = 0.005 \) for each trial.
(ii) The probability that a particular employee receives at least one gift card is:
\( P(X \geq 1) = 1 – P(X = 0) \)
\( P(X = 0) = \binom{52}{0}(0.005)^0(0.995)^{52} = (1)(1)(0.995)^{52} \approx 0.7705 \)
\( P(X \geq 1) = 1 – 0.7705 = 0.2295 \)
The probability is approximately \( \boxed{0.2295} \).
(b)
The expected value is:
\( E(X) = np = 52 \times 0.005 = 0.26 \)
Interpretation: If this random selection process is repeated over many years, a particular employee would receive an average of \( 0.26 \) gift cards per year, or about one gift card every four years.
(c)
No, Agatha does not have a strong argument that the selection process was not truly random.
Explanation: The probability that a particular employee receives no gift cards in a \( 52 \)-week year is \( (0.995)^{52} \approx 0.7705 \), which means there’s about a \( 77\% \) chance that any given employee would not receive a gift card in a year. Since this is quite likely to occur by random chance, Agatha’s experience does not provide strong evidence against the randomness of the selection process.