Answer: Probability that Charles gets 7 or more correctly = 0.776
Probability that Ruth wins = 0.827
Step-by-step explanation:
Given data:
Probability that Charles distinguishes between beer and ale = 0.75
no of glasses = 10
no required to win the bet = 7 or more.
Solution:
a) probability of success p=0.75.
This is a binomial probability problem, what we need to solve for is the probability that ( X >=7 ) this can be gotten as
P(X>=7) = (10Ck) (0.75)^k (1-0.75)^(10-k)
where k=7,8,9,10
Probability that he gets 7 correctly.
P(X>=7) = (10Ck) (0.75)^k (1-0.75)^(10-k)
Substitute k = 7 into the equation
= 0.250
Probability that he gets 8 correctly.
P(X>=7) = (10Ck) (0.75)^k (1-0.75)^(10-k)
substitute k = 8 into the equation
= 0.282
Probability that he gets 9 correctly.
P(X>=7) = (10Ck) (0.75)^k (1-0.75)^(10-k)
Substitute k = 9 into the equation
= 0.188
Probability that he gets 10 correctly.
P(X>=7) = (10Ck) (0.75)^k (1-0.75)^(10-k)
Substitute the k= 10 into the equation.
= 0.0563
Probability that Charles gets 7 or more correctly
= 0.250 + 0.282 + 0.0563 + 0.188
= 0.776
b) Probability that Ruth wins
K = 1,2,3,4,5,6
p=0.5
P(X < 7)
P(X < 7) = (10Ck) (.5)^k (1-.5)^(10-k)
Probability that he gets one correctly
P(X = 1 ) = (10Ck) (.5)^k (1-.5)^(10-k)
substitute k= 1 into the equation.
= 0.0098
P(X=2) = (10Ck) (.5)^k (1-.5)^(10-k)
Substitute k= 2 into the equation
= 0.0439
P(X =3) = (10Ck) (.5)^k (1-.5)^(10-k)
Substitute k=3 into the equation.
=0.117
P(X=4) = (10Ck) (.5)^k (1-.5)^(10-k)
Substitute k= 4 into the equation
= 0.2051
P(X=5) = (10Ck) (.5)^k (1-.5)^(10-k)
Substitute k =5 into the equation
= 0.246
P(X= 6 ) = (10Ck) (.5)^k (1-.5)^(10-k)
Substitute k = 6 into the equation
= 0.205
Probability that Ruth wins
= 0.205 + 0.246 + 0.2051 + 0.117 + 0.0439 + 0.0098
= 0.827
