Question

a certain game consists of rolling a single fair die and pays off as follows: $5 for a 6, $2 for a 5, $1 for a 4, and no payoff otherwise. find the expected winnings for this game.

Answer 1

I'm assuming you don't pay to play this game right?

Answer 2

exactly

Answer 3

if so, then

P(rolling 6) = 1/6
W(rolling 6) = 5

Note: W(rolling 6) = "amount of money you win for rolling a six...so it's your winnings if you roll a 6"

P(rolling 5) = 1/6
W(rolling 5) = 2

P(rolling 4) = 1/6
W(rolling 4) = 1

P(rolling 3,2,or1) = 3/6
W(rolling 3,2,or1) = 0

Answer 4

Expected Value

E[X] = P(rolling 6)*W(rolling 6)+P(rolling 5)*W(rolling 5)+P(rolling 4)*W(rolling 4)+P(rolling 3,2,1)*W(rolling 3,2,1)

E[X] = (1/6)*(5)+(1/6)*(2)+(1/6)*(1)+(3/6)*(0)

E[X] = 5/6 + 2/6 + 1/6 + 0/6

E[X] = (5 + 2 + 1 + 0)/6

E[X] = 8/6

E[X] = 4/3

E[X] = 1.33333333...

E[X] = 1.33

Answer 5

So assuming you do not pay to play, you would expect to earn $1.33 for each roll (on average)