Question

Suppose it costs $41 to roll a pair of dice. You get paid $6 times thesum of the numbers that appear on the dice. Is it a fair game?

Answer 1

we know that

the total possible outcomes=6*6=36

the possible outcomes are

1 1 ----> sum 2 ---> P=1/36

1 2 1 3 14 15 16 ---> sum 3 4 5 6 7 ----> P=1/36 -----> total sum=25

2 1 22 23 24 25 26 ----> sum 3 4 5 6 7 8 ----> P=1/36 ----> total sum=33

3 1 3 2 3 3 34 35 36 ----> sum 4 5 6 7 8 9 ---> P=1/36 ----> total sum=39

4 1 42 43 44 45 46 ----> sum 5 6 7 8 9 10 ---> P=1/36 ---> total sum=45

51 52 53 54 55 56 ---> sum 6 7 8 9 10 11 ---> P=1/6 -----> total sum=51

61 62 63 64 65 66 ---> sum 7 8 9 10 11 12 --->P=1/36 ----> total sum=57

Find out the expected value

EV=(1/36)[2+25+33+39+45+51+57]*6

EV=(1/36){252]*6

EV=$7*6=$42

$42>$41

therefore

The answer is

Yes, because the expected value of the game without considering the cost is $42