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AP Statistics 4.8 Mean and Standard Deviation of Random Variables- FRQs - Exam Style Questions

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Question

In an online game, players move through a virtual world collecting geodes, a type of hollow rock. When broken open, these geodes contain crystals of different colors that are useful in the game. A red crystal is the most useful crystal in the game. The color of the crystal in each geode is independent and the probability that a geode contains a red crystal is \(0.08\).
(a) Sarah, a player, will collect and open geodes until a red crystal is found.
(i) Calculate the mean of the distribution of the number of geodes Sarah will open until a red crystal is found. Show your work.
(ii) Calculate the standard deviation of the distribution of the number of geodes Sarah will open until a red crystal is found. Show your work.

(b) Another player, Conrad, decides to play the game and will stop opening geodes after finding a red crystal or when \(4\) geodes have been opened, whichever comes first. Let \(Y=\) the number of geodes Conrad will open. The table shows the partially completed probability distribution for the random variable Y.

Number of geodes Conrad will open, y\(1\)\(2\)\(3\)\(4\)
Probability, P(Y=y)\(0.08\)\(0.0736\)  

(i) Calculate \(P(Y=3)\). Show your work.
(ii) Calculate \(P(Y=4)\). Show your work.

(c) Consider the table and your results from part (b).
(i) Calculate the mean of the distribution of the number of geodes Conrad will open. Show your work.
(ii) Interpret the mean of the distribution of the number of geodes Conrad will open, which was calculated in part (c-i).

Most-appropriate topic codes (CED):

• TOPIC 4.12: The Geometric Distribution
• TOPIC 4.7: Introduction to Random Variables and Probability Distributions
• TOPIC 4.8: Mean and Standard Deviation of Random Variables
▶️ Answer/Explanation
Detailed solution

(a)
The number of geodes Sarah opens, G, follows a geometric distribution with probability of success \(p=0.08\).
(i) The mean of a geometric distribution is \(\mu = \frac{1}{p}\).
\(\mu = \frac{1}{0.08} = 12.5\) geodes.
(ii) The standard deviation of a geometric distribution is \(\sigma = \frac{\sqrt{1-p}}{p}\).
\(\sigma = \frac{\sqrt{1-0.08}}{0.08} = \frac{\sqrt{0.92}}{0.08} \approx 11.99\) geodes.

(b)
(i) \(P(Y=3)\) is the probability that the first red crystal is found on the third geode. This means the first two are not red and the third is red.
\(P(Y=3) = (1-0.08)^2 (0.08) = (0.92)^2 (0.08) \approx 0.0677\).
(ii) Conrad stops at \(4\) geodes if he does not find a red crystal in the first three tries. The outcome of the fourth geode does not matter. The probability of this is the sum of the probabilities of all outcomes that result in stopping at \(Y=4\). The easiest way to calculate this is to use the complement rule, as the probabilities for \(Y=1, 2, 3, 4\) must sum to \(1\).
\(P(Y=4) = 1 – [P(Y=1) + P(Y=2) + P(Y=3)]\)
\(P(Y=4) \approx 1 – (0.08 + 0.0736 + 0.0677) = 1 – 0.2213 = 0.7787\).

(c)
(i) The mean of the discrete random variable Y is calculated as \(E(Y) = \sum y \cdot P(Y=y)\).
\(E(Y) \approx (1)(0.08) + (2)(0.0736) + (3)(0.0677) + (4)(0.7787)\)
\(E(Y) \approx 0.08 + 0.1472 + 0.2031 + 3.1148 \approx 3.545\) geodes.
(ii) The mean of approximately \(3.545\) geodes is the long-run average number of geodes Conrad would open per attempt if he were to repeat this process many, many times.

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