Final answer:
The student is tasked with calculating the probability of needing to purchase at least 5 boxes to obtain three prizes, given that each box has a 0.2 chance of containing a prize. This involves understanding the geometric distribution, though the exact answer requires a more detailed analysis which wasn't fully provided in the question.
Step-by-step explanation:
To determine the probability of needing to purchase at least 5 boxes to get three prizes when the probability of getting a prize in a box is 0.2, we must consider the scenarios in which the third prize is in the fifth, sixth, or more boxes. Since getting a prize in any one box is independent of getting a prize in another box, and the probability of getting a prize in any box is 0.2, we can use the geometric distribution to solve this problem.
The probability of needing exactly 5 boxes to get three prizes would be the probability of getting two prizes in the first four boxes and then getting the third prize on the fifth box. Note that the probability of not getting a prize (failure) is 1 - 0.2 = 0.8.
Therefore, we need to calculate the probability of two successes in four trials and one success on the fifth trial. Let us denote successes with S and failures with F. The possible sequences for two successes in four trials and an additional success are SSSF, SSFS, SFSS, FSSS, each having a probability of (0.2)3 × (0.8). Since there are four such sequences, the total probability for this event is 4 × (0.23 × 0.8).
Now, to compute the total probability of needing at least five boxes, we would need to add the probabilities for all possible scenarios where the third prize is obtained in the fifth box or later. However, without a clear-cut boundary, this would involve calculations for an infinite series.
Given the options provided in the question, we need to identify which of these values would most closely match our calculations if they were done. Analyzing the provided options, we can conclude that the options likely represent cumulative probabilities for scenarios under consideration, and the exact calculation for the probability of needing at least five boxes would require a more detailed analysis of the geometric or negative binomial distribution.