If multiple collectors are competing to complete a set of coupons, what is the expected time it takes for the first collector to complete their set? How does this change the dynamics of the problem compared to the single collector case?
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In the case where multiple collectors are competing to complete a set of coupons, the expected time it takes for the first collector to complete their set is reduced compared to the single collector case.
Let's consider the single collector case first. If there is only one collector, each coupon has an equal probability of being drawn by the collector in each attempt. The expected time for the collector to complete the set is determined by the coupon rarity and follows the classic Coupon Collector's Problem, which has an expected time of approximately n * ln(n) attempts, where n is the total number of coupons in the set.
Now, let's introduce multiple collectors. Each collector independently draws coupons randomly. As there are multiple collectors, the probability of any given coupon being drawn by at least one collector increases with each attempt. This increased probability of a coupon being drawn reduces the expected time for the first collector to complete their set.
To calculate the expected time with multiple collectors, the exact dynamics of the problem need to be defined. Factors like the number of collectors, probability distribution of drawing coupons, and the rules governing the sharing or trading of coupons among collectors need to be considered.
For example, if there are R collectors and each collector randomly draws a coupon with equal probability in each attempt, then the probability of any specific collector drawing a new coupon is (n-C)/(n), where n is the total number of coupons and C is the number of coupons already collected by that specific collector. With R collectors drawing independently, the probability of at least one collector drawing a new coupon in a single attempt is (n-C)^(R)/n^(R).
Using this probability, the expected time for the first collector to complete their set can be approximated. However, the exact calculation can become complex and depend on various assumptions about the collector behavior.
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