Final answer:
The probability of finding a prize at most in three cereal boxes combines the individual probabilities of finding a prize in one, two, or three attempts, while the probability of exactly three boxes is a specific case of the third attempt. More than three boxes is calculated as the complement of the cumulative probability of the first three.
Step-by-step explanation:
The probability that at most three cereal boxes must be purchased to find a prize is calculated by considering the probability of finding a prize in one, two, or three boxes. Since 7% of cereal boxes contain a prize, the probability of not finding a prize in any given box is 93%. Therefore, the probability of finding one on the first purchase is 0.07. If it's not found on the first box, then the probability of finding it on the second box is 0.93 × 0.07. Finally, if not found in the first two, then the probability of finding it on the third purchase is 0.93 × 0.93 × 0.07. To find the total probability of at most three purchases, we add these probabilities together.
For exactly three boxes, we use the scenario where the prize is not in the first two boxes but is in the third, which we already determined is 0.93× 0.93 × 0.07.
To find the probability that more than three boxes must be purchased, we consider the complement of the probability of finding a prize in up to three boxes, which we calculate as 1 minus the probability of finding a prize in at most three boxes.