Question

A certain game consists of rolling a single fair die and pays off as​ follows: ​$6 for a​ 6, ​$4 for a​ 5, ​$1 for a​ 4, and no payoff otherwise. Find the expected winnings for this game

Answer 1

Let X is a random variable shows the winning amount. Here X can take values $6, $4, $1 and $0.

When a single fair die is rolled the possible outcomes are 1, 2, 3, 4, 5, and 6.

Out of 6 outcomes, 1 is 6 so when X= $6 then

P(X=$6) = 1/6

Out of 6 outcomes, 1 is 5 so when X= $5 then

P(X=$5) = 1/6

Out of 6 outcomes, 1 is 4 so when X= $1 then

P(X=$1) = 1/6

And

P(X=$0) = 1- P(X=$6) - P(X=$5) - P(X=$1) = 3/6

Following table shows calculations for expected winning:

X	P(X=x)	xP(X=x)
6	1/6	6/6
5	1/6	5/6
1	1/6	1/6
0	3/6	0
Total		2

So expected winning is