Probability and Random Variables: AP Statistics Math Test Answer Key – PDF Download

AP Statistics Probability Test Answer Key: questions 1–5

Question 1: Gym members (3 points)

1. Joint: \(\frac{90}{300}=0.30\).
2. Marginal: \(\frac{105}{300}=0.35\).
3. Conditional on the 60 class members: \(\frac{30}{60}=0.50\).

Scoring: 1 point each. (a) 0.30 from 90 over 300. (b) 0.35 from the column total. (c) 0.50, with denominator 60 and not 300.

Question 2: Time and activity (2 points)

1. \(P(\text{weights})=\frac{135}{300}=0.45\) and \(P(\text{weights}|\text{morning})=\frac{45}{135}\approx 0.333\).
2. The probabilities differ (0.45 versus 0.333), so the events are not independent: knowing a member visits in the morning changes the probability that the member lifts weights.

Scoring: (a) 1 point for both values. (b) 1 point for “not independent” with a comparison of the two probabilities.

Question 3: Bakery customers (2 points)

1. \(P(\text{both})=0.55+0.30-0.70=0.15\).
2. Since \(0.15\neq 0\), the events are not mutually exclusive. Since \(0.55\times 0.30=0.165\neq 0.15\), they are not independent.

Scoring: (a) 1 point. (b) 1 point for both conclusions with the supporting calculation.

Question 4: Late bus (3 points)

1. \(P(\text{rain and late})=0.3\times 0.2=0.06\).
2. \(P(\text{late})=0.06+0.7\times 0.05=0.06+0.035=0.095\).
3. \(P(\text{rain}|\text{late})=\frac{0.06}{0.095}\approx 0.632\).

Scoring: (a) 1 point for the product. (b) 1 point for adding both late branches. (c) 1 point for conditional probability as a ratio, 0.632.

Question 5: Guessing on a quiz (2 points)

1. Let digits 0 and 1 mean a correct guess (2 of 10 digits = 20%) and 2 to 9 an incorrect guess. Read 4 digits per trial and count the correct guesses; repeat many times .
2. At least 2 correct occurred in \(6+1+0=7\) of 40 trials: \(\frac{7}{40}=0.175\).

Scoring: (a) 1 point for a digit assignment with the correct 20% and 4 digits per trial. (b) 1 point for 7 out of 40 = 0.175.

AP Statistics Probability Test Answer Key: questions 6–8

Question 6: Online orders (3 points)

1. \(0.10+0.25+0.35+k+0.10=1\), so \(k=0.20\).
2. \(\mu_X=0+0.25+0.70+0.60+0.40=1.95\) orders.
3. \(\sigma_X^{2}=(1.95)^{2}(0.10)+(0.95)^{2}(0.25)+(0.05)^{2}(0.35)+(1.05)^{2}(0.20)+(2.05)^{2}(0.10)=0.38025+0.225625+0.000875+0.2205+0.42025=1.2475\), so \(\sigma_X\approx 1.12\) orders.

Scoring: 1 point each. (a) 0.20. (b) 1.95 with the sum of products. (c) 1.12, with correct weighted squared deviations.

Question 7: Carnival wheel (2 points)

1. \(\mu=20\cdot\frac{1}{8}+5\cdot\frac{2}{8}+0\cdot\frac{5}{8}=2.50+1.25=3.75\) dollars.
2. \(3.75-4=-\$0.25\). Over many spins, a player loses on average $0.25 per spin.

Scoring: (a) 1 point for 3.75. (b) 1 point for −$0.25 with an interpretation as a long-run average per spin.

Question 8: Socks (3 points)

1. \(\frac{6}{10}\cdot\frac{5}{9}=\frac{30}{90}=\frac{1}{3}\).
2. \(\frac{6}{10}\cdot\frac{4}{9}+\frac{4}{10}\cdot\frac{6}{9}=\frac{24}{90}+\frac{24}{90}=\frac{48}{90}=\frac{8}{15}\approx 0.533\).
3. \(P(\text{second black})=\frac{6}{10}=0.6\) but \(P(\text{second black}|\text{first black})=\frac{5}{9}\approx 0.556\). The probabilities differ, so the events are not independent.

Scoring: (a) 1 point for 1/3. (b) 1 point for adding both orders, 8/15. (c) 1 point for comparing the two probabilities and concluding not independent.

Scoring summary

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