Answer to Question #91984 in Statistics and Probability for mki

Question

An urn contains 5 balls labeled 1, 2, …, 5. From the urn, 6 balls are drawn with replacement and the numbers on the balls drawn are noted down on six tokens (a separate token for each draw). All six tokens are placed in a jar. What is the probability that all numbers on tokens in the jar are the same? Consider the case of sampling with replacement without ordering.

Answer 1

The total number of ways to select 6 balls out of 5 with replacement is 5⁶.

All numbers on tokens are the same if and only if all selected balls have the same number,

there are 5 possibilities {1, 1, 1, 1, 1, 1}, {2, 2, 2, 2, 2, 2},..., {5, 5, 5, 5, 5, 5} to select balls with the same number, therefor the probability that all numbers on tokens in the jar are the same is 5/5⁶ = 1/5⁵ = 0.00032.

(In the case of sampling with replacement without ordering not all sequence of numbers in the jar have the same probability, for example there is only one way to select balls corresponding to the sequence {1, 1, 1, 1, 1, 1}, and 6 ways to select balls corresponding to the sequence {2, 1, 1, 1, 1, 1}.)

There are C(n + k - 1, k) k-combinations with repetitions from a set with n distinct elements (where C(m, l) is binomial coefficient), so there are C(5 + 6 - 1, 6) = C(10, 6) = 10!/(6!4!) = 210 combinations of numbers in the jar. If all unordered combinations have the same probability, then the probability that all numbers on tokens in the jar are the same is 5/210 = 1/42.

Answer: In the case of sampling with replacement with ordering the probability that all numbers on tokens in the jar are the same is 1/5⁵ = 0.00032,

In the case of sampling with replacement without ordering the probability that all numbers on tokens in the jar are the same is 5/210 = 1/42.