Coupon Collector Problem: Probability of Collecting All 108 Types

This article discusses the probability of collecting all 108 different types of coupons when buying 'n' boxes of instant noodles, assuming each box contains one random coupon. This is a classic example of the 'coupon collector's problem' in probability theory.

The problem aims to determine the probability of collecting all 108 coupons when purchasing 'n' boxes, where 'n' is greater than or equal to 200.

The probability of not getting a new coupon after buying a box, given that we already have 'x' unique coupons, is (108 - x) / 108. Therefore, the probability of not getting a new coupon in 'n' boxes is ((108 - x) / 108)^n.

To find the probability of successfully collecting all coupons, we calculate the complement of not getting a new coupon. So, the probability of collecting all 108 coupons is 1 - ((108 - x) / 108)^n.

In our case, we want to calculate the probability when 'n' is greater than or equal to 200 and we haven't collected any coupons yet (x = 0). The formula then becomes:

Probability = 1 - ((108 - 0) / 108)^n

  = 1 - (1 / 108)^n  

Therefore, the probability of collecting all 108 coupons is 1 - (1 / 108)^n, where 'n' is greater than or equal to 200.

It's important to note that this probability is an approximation. It assumes that each coupon has an equal chance of appearing and doesn't consider the possibility of receiving duplicates. In real-world scenarios, the probability might vary.