Question

A game of chance involves rolling an unevenly balanced 4-sided die. The probability that a roll comes up 1 is 0.22, the probability that a roll comes up 1 or 2 is 0.42, and the probability that a roll comes up 2 or 3 is 0.54 . If you win the amount that appears on the die, what is your expected winnings? (Note that the die has 4 sides.)

Answer 1

Expected Winnings = 2.6

Explanation:

Since the probability of rolling a 1 is 0.22 and the probability of rolling either a 1 or a 2 is 0.42, the probability of rolling only a 2 can be determined as:

`P_(1,2) = P_(1)+P_(2)\\P_(2) = 0.42 - 0.22 = 0.20`

The same logic can be applied to find the probability of rolling a 3

`P_(2,3) = P_(2)+P_(3)\\P_(3) = 0.54 - 0.20 = 0.34`

The sum of all probabilities must equal 1.00, so the probability of rolling a 4 is:

`P_(4) =1- P_(1)+P_(2)+P_(3) = 1-0.22+0.20+0.34\\P_(4)=0.24`

The expected winnings (EW) is found by adding the product of each value by its likelihood:

`EW=1*P_(1)+2*P_(2)+ 3*P_(3)+ 4*P_(4) \\EW=1*0.22+2*0.20+ 3*0.34+ 4*0.24\\EW=2.6`

Expected Winnings = 2.6