Question

At a carnival game, Katya chooses a number from 1 through 6 and rolls two ordinary dice. If her number comes up on one die, she wins three tickets. If her number comes up on both dice, she wins six tickets. If she plays all morning, how many tickets would she expect to win, on average, per turn?

Answer 1

1 dollar

Explanation:

Given that at a carnival game, Katya chooses a number from 1 through 6 and rolls two ordinary dice.

If her number comes up on one die, she wins three tickets. If her number comes up on both dice

She can select any number from 1 to 6.

Getting her number in one die is having p = 1/6

X - no of dice she gets that number is binomial with n =2 and p =1/6

The probability for getting one number she chose in only one die =
`2C1(1)/(6) (5)/(6)\\=(5)/(18)`

Probability for getting the chosen number on both dice =
`(1)/(6)*(1)/(6)\\=(1)/(36)`

Her gain is

Y 3 6 0

Prob 5/18 1/36 25/36

E(Y) = expected value to win

=
`(15)/(18)+6*(1)/(36)\\=1`

Her expected gain in one turn is exactly 1 dollar