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AP Statistics 4.12 The Geometric Distribution- MCQs - Exam Style Questions

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Question

A club will sell candy bars. Some bars have tickets under the wrapper that can be redeemed for a prize; the probability a randomly selected candy bar contains a ticket is \(p=0.11\). Mara will purchase and unwrap one bar at a time until she finds a ticket.
Which of the following is closest to the probability that Mara will purchase four or fewer candy bars?
(A) 0.0001
(B) 0.0012
(C) 0.0775
(D) 0.3726
(E) 0.6274
▶️ Answer/Explanation
Detailed solution
Let \(X\sim\text{Geom}(p=0.11)\) be the number of bars until the first ticket (includes the success).
We want \(P(X\le4)=1-P(X\ge5)=1-(1-p)^4\).
Compute: \(1-p=0.89\).
\(0.89^2=0.7921\) and \(0.89^4=(0.7921)^2\approx 0.6274\).
Therefore \(P(X\le4)=1-0.6274=0.3726\).
✅ Answer: (D) 0.3726

Question

A die used in a certain board game has eight faces, of which \(3\) are red, \(3\) are yellow, and \(2\) are blue. Each face is equally likely to land faceup when the die is tossed. In the game, a player tosses the die until blue lands faceup, and the number of tosses before blue lands faceup is counted. For example, a player who tosses the sequence shown in the following table has tossed the die \(3\) times before blue lands faceup.
Toss number\(1\)\(2\)\(3\)\(4\)
Face coloryellowyellowredblue
What is the probability that a player will toss the die at least \(2\) times before blue lands faceup?
(A) \(0.1406\)
(B) \(0.4219\)
(C) \(0.4375\)
(D) \(0.5625\)
(E) \(0.5781\)
▶️ Answer/Explanation
Detailed solution

Let \(B\) be the number of tosses before the first blue. Each toss has \(P(\text{blue})=2/8=0.25\) and \(P(\text{nonblue})=0.75\). Thus \(B\) is geometric with \(p=0.25\). We need \[ P(B\ge 2)=1-P(B<2)=1-[P(B=0)+P(B=1)]. \] Here \(P(B=0)=0.25\) (blue on first toss) and \(P(B=1)=0.75(0.25)=0.1875\). Therefore \[ P(B\ge 2)=1-(0.25+0.1875)=1-0.4375=0.5625 . \] ✅ Answer: (D)

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