Homework Answers
Probability that a person playing one round of the game wins =
P(the person wins with first outcome 7) + P(the person wins with first outcome 11) + P(the person wins with first outcome 4) + P(the person wins with first outcome 5) + P(the person wins with first outcome 6) + P(the person wins with first outcome 8) + P(the person wins with first outcome 9) + P(the person wins with first outcome 10)
.
P(first outcome is 7) = P(1,6)+P(6,1)+P(2,5)+P(5,2)+P(3,4)+P(4,3) = 6/36 = 1/6
P(first outcome is 11) = P(5,6)+P(6,5) = 2/36 = 1/18
P(first outcome is 4) = P(1,3)+P(3,1)+P(2,2) = 3/36 = 1/12
P(first outcome is 5) = P(1,4)+P(4,1)+P(2,3)+P(3,2) = 4/36 = 1/9
P(first outcome is 6) = P(1,5)+P(5,1)+P(2,4)+P(4,2)+P(3,3) = 5/36
P(first outcome is 8) = P(2,6)+P(6,2)+P(3,5)+P(5,3)+P(4,4) = 5/36
P(first outcome is 9) = P(3,6)+P(6,3)+P(4,5)+P(5,4) = 4/36 = 1/9
P(first outcome is 10) = P(4,6)+P(6,4)+P(5,5) = 3/36 = 1/12
.
P(the person wins with first outcome 7) = P(first outcome is 7) = 1/6
P(the person wins with first outcome 11) = P(first outcome is 11) = 1/18
P(the person wins with first outcome 4)
= P(first outcome is 4 , second outcome is 4) + P(first outcome is 4 , second outcome is not 7 , third outcome is 4) + P(first outcome is 4 , second and third outcome is not 4 , fourth outcome is 4) + ........
= (1/12)(1/12) + (1/12)(1-1/6)(1/12) + (1/12)(1-1/6)2(1/12) + .........
= (1/12)2 + (1/12)2(5/6) + (1/12)2(5/6)2 + .........
P(the person wins with first outcome 5)
= P(first outcome is 5 , second outcome is 5) + P(first outcome is 5 , second outcome is not 7 , third outcome is 5) + P(first outcome is , second and third outcome is not 4 , fourth outcome is 5) + ........
= (1/9)(1/9) + (1/9)(1-1/6)(1/9) + (1/9)(1-1/6)2(1/9) + .........
= (1/9)2 + (1/9)2(5/6) + (1/9)2(5/6)2 + .........
P(the person wins with first outcome 6)
= P(first outcome is 6 , second outcome is 6) + P(first outcome is 6 , second outcome is not 7 , third outcome is 6) + P(first outcome is 6 , second and third outcome is not 4 , fourth outcome is 6) + ........
= (5/36)(5/36) + (5/36)(1-1/6)(5/36) + (5/36)(1-1/6)2(5/36) + .........
= (5/36)2 + (5/36)2(5/6) + (5/36)2(5/6)2 + .........
P(the person wins with first outcome 8)
= P(first outcome is 8 , second outcome is 8) + P(first outcome is 8 , second outcome is not 7 , third outcome is 8) + P(first outcome is 8 , second and third outcome is not 4 , fourth outcome is 8) + ........
= (5/36)(5/36) + (5/36)(1-1/6)(5/36) + (5/36)(1-1/6)2(5/36) + .........
= (5/36)2 + (5/36)2(5/6) + (5/36)2(5/6)2 + .........
P(the person wins with first outcome 9)
= P(first outcome is 9 , second outcome is 9) + P(first outcome is 9 , second outcome is not 7 , third outcome is 9) + P(first outcome is 9 , second and third outcome is not 4 , fourth outcome is 9) + ........
= (1/9)(1/9) + (1/9)(1-1/6)(1/9) + (1/9)(1-1/6)2(1/9) + .........
= (1/9)2 + (1/9)2(5/6) + (1/9)2(5/6)2 + .........
P(the person wins with first outcome 10)
= P(first outcome is 10 , second outcome is 10) + P(first outcome is 10 , second outcome is not 7 , third outcome is 10) + P(first outcome is 10 , second and third outcome is not 4 , fourth outcome is 10) + ........
= (1/12)(1/12) + (1/12)(1-1/6)(1/12) + (1/12)(1-1/6)2(1/12) + .........
= (1/12)2 + (1/12)2(5/6) + (1/12)2(5/6)2 + .........
.
Probability
that a person playing one round of the game wins
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