Homework SourseAfter paying 5 a player throws simultaneously a fair coin If
After paying $5, a player throws simultaneously a fair coin. If the coin lands on heads, then he rolls a fair die and he wins the vale that appears on the die. If the coin lands on tails, then he rolls two fair die and he wins the sum of the vale on the two dice. Find the expected value of the player\'s net winnings.
The answer is 1/4 but I don\'t know how to find it.
Solution
After paying $5, a player throws simultaneously a fair coin. If the coin lands on heads, then he rolls a fair die and he wins the vale that appears on the die. If the coin lands on tails, then he rolls two fair die and he wins the sum of the vale on the two dice. Find the expected value of the player\'s net winnings.
Probability of head =0.5
Expected value of single die = Expectation = E(x) = [x·P(x)] = (1/6)(1+2+3+4+5+6) =3.5
Expected earning =0.5*3.5=1.75
Probability of tail =0.5
Expectation of sum:
Here are all 36 possible rolls with a pair of dice:
(1,1) (1,2) (1,3) (1,4) (1,5) (1,6)
(2,1) (2,2) (2,3) (2,4) (2,5) (2,6)
(3,1) (3,2) (3,3) (3,4) (3,5) (3,6)
(4,1) (4,2) (4,3) (4,4) (4,5) (4,6)
(5,1) (5,2) (5,3) (5,4) (5,5) (5,6)
(6,1) (6,2) (6,3) (6,4) (6,5) (6,6)
Their sums are
2 3 4 5 6 7
3 4 5 6 7 8
4 5 6 7 8 9
5 6 7 8 9 10
6 7 8 9 10 11
7 8 9 10 11 12
Sum of roll Prob. of sum
x P(x) x·P(x)
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2 1/36 2/36
3 2/36 6/36
4 3/36 12/36
5 4/36 20/36
6 5/36 30/36
7 6/36 42/36
8 5/36 40/36
9 4/36 36/36
10 3/36 30/36
11 2/36 22/36
12 1/36 12/36
-------------------------------------
TOTALS 36/36=1 252/36 = 7
Expectation = E(x) = [x·P(x)] = 7
Expected earning =0.5*7=3.5
Total expected value =1.75+3.5=$5.25
expected value of the player\'s net winnings = 5.25 - 5 = $ 0.25
