A box contains five slips of paper, marked $1, $1, $10, $10, and $25. The winner of a contest selects two slips of paper at random and then gets the larger of the dollar amounts on the two slips. Define a random variable w by w = amount awarded. Determine the probability distribution of w. (Hint: Think of the slips as numbered 1, 2, 3, 4, and 5, so that an outcome of the experiment consists of two of these numbers.)

P (w = $1) = 1/10

P (w = $10) = 5/10

P (w = $25) = 4/10

Step-by-step explanation:

Let S1, S2, S3, S4 and S5 be the slips of paper, where we make the next identifications

S1: $1, S2: $1, S3: $10, S4: $10 and S5: $25. The number of different subsets of size 2 that we can get with the five slips of paper is given by 5C2 = 10 (combinations). Let's define the following events

A: the winner of the contest gets $1

B: the winner of the contest gets $10

C: the winner of the contest gets $25

So,

A={ (S1, S2) }

B={ (S1, S3), (S2, S3), (S1, S4), (S2, S4), (S3, S4) }

C={ (S1, S5), (S2, S5), (S3, S5), (S4, S5) }

The random variable w can only take the values $1, $10 or $25. Then

P (w = $1) = P (A) = 1/10

P (w = $10) = P (B) = 5/10

P (w = $25) = P (C) = 4/10