4. Charles claims that he can distinguish between beer and ale 75 percent of the time. Ruth bets that he cannot and, in fact, just guesses. To settle this, a bet is made: Charles is to be given ten small glasses, each having been filled with beer or ale, chosen by tossing a fair coin. He wins the bet if he gets seven or more correct. Find the probability that Charles wins if he has the ability that he claims. Find the probability that Ruth wins if Charles is guessing.

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

if x = 1, then the answer is 3.5

answer: both chords and diameters have two endpoints on a circle. diameters must intersect the center of a circle.