Question

Suppose that \(5 \%\) of cereal boxes contain a prize and the other \(95 \%\) contain the message, "Sorry, try again." Consider the random variable \(x,\) where \(x=\) number of boxes purchased until a prize is found. a. What is the probability that at most two boxes must be purchased? b. What is the probability that exactly four boxes must be purchased? c. What is the probability that more than four boxes must be purchased?

Short answer

a. The probability that at most two boxes must be purchased is 0.0975 or 9.75 %. \n b. The probability that exactly four boxes must be purchased is 0.043125 or 4.3125 %. \n c. The probability that more than four boxes must be purchased is 0.815625 or 81.5625 %.

Step-by-step solution

Determining the Probability of Success

The probability of success, \(p\), for this scenario, which is finding a prize in a cereal box, is defined as \(0.05\) or \(5\% \).

Calculating the Probability for at Most Two Purchases

It is needed to calculate the probability of finding a prize within 1 or 2 purchases. So, it is the sum of the probabilities for x = 1 and x = 2.

Execute Calculation for Step 2

Substitute x = 1 and x = 2 into the geometric distribution formula and sum the results: \(P(x \leq 2) = P(x=1) + P(x=2) = (1 - 0.05)^{(1-1)} * 0.05 + (1 - 0.05)^{(2-1)} * 0.05 = 0.05 + 0.0475 = 0.0975.\)

Calculating the Probability for Exactly Four Purchases

Find the probability of finding a prize exactly on the fourth purchase. Therefore, substitute x = 4 into the geometric distribution formula.

Execute Calculation for Step 4

Applying the formula \(P(x=4) = (1 - 0.05)^{(4-1)} * 0.05 = 0.043125.\)

Calculating the Probability for More Than Four Purchases

To determine the probability of the first success taking place after more than four trials, it is necessary to find the probability of the first success happening in at most four trials (from step 2 and step 4) and then subtracting this from 1.

Execute Calculation for Step 6

So, \(P(x > 4) = 1 - P(x \leq 4) = 1 - ( P(x=1) + P(x=2) + P(x=3) + P(x=4) ) = 1 - ( P(x \leq 2) + P(x=3) + P(x=4) ) = 1 - (0.0975 + (1 - 0.05)^{(3-1)} * 0.05 + 0.043125) = 0.815625.\)