91Ó°ÊÓ

First, understand the problem by rewritting it in terms of conditional probability. If Tommy scores a shot, the probability of scoring the next shot, P(S | S), is twice the probability of missing the next shot, P(M | S). Meanwhile, if Tommy misses a shot, the probability of missing the next shot, P(M | M), is three times the probability of scoring on the next shot, P(S | M).

To find the probability that Tommy will score two shots later after missing a shot, we will analyze all the possibilities using a probability tree. The probability we are looking for is P(S | M) * P(S | S) + P(M | M) * P(S | M). We know that: 1. P(M | S) : P(S | S) = 1:2 2. P(S | M) : P(M | M) = 1:3 Using these ratios, we can calculate the conditional probabilities: 1. P(S | S) = 2/3 ; P(M | S) = 1/3 2. P(S | M) = 1/4 ; P(M | M) = 3/4 Now we can calculate the probability that Tommy scores two shots later after missing a shot as: P(S after 2 shots | M) = P(S | M) * P(S | S) + P(M | M) * P(S | M) = (1/4)*(2/3) + (3/4)*(1/4) = 1/6 + 3/16 = 9/16

We will now set up a system of equations with the following variables: x = P(Tommy scores a shot) y = P(Tommy misses a shot) We have the following equations: x = P(S | S) * x + P(S | M) * y y = P(M | S) * x + P(M | M) * y Substituting our conditional probabilities, we get: x = (2/3)*x + (1/4)*y y = (1/3)*x + (3/4)*y Now we solve the system of equations for x and y: x = (2/3)*x + (1/4)*y (1/3)*x = (1/4)*y y = 4x/3 Substitute y in the second equation: y = (1/3)*x + (3/4)*y 4x/3 = (1/3)*x + (3/4)*(4x/3) 4x/3 = (1/3)*x + x (1/3)*x = -1/3*x x = 1/2 Now find y using x: y = 4x/3 y = 4(1/2)/3 y = 4/6 y = 2/3 Tommy's long-term percentage of successful shots is x*100 = 1/2 * 100 = 50%.

a. The probability that Tommy will score two shots later after missing a shot is 9/16. b. In the long term, Tommy's percentage of successful shots is 50%.