Question

Winning at tennis A player serving in tennis has two chances to get a serve into play. If the first serve goes out of bounds, the player serves again. If the second serve is also out, the player loses the point. Here are probabilities based on four years of the Wimbledon Championship:18 P(1st serve in) = $0.59$ P(win point | 1st serve in) = $0.73$ P(2nd serve in | 1st serve out) = $0.86$P(win point | 1st serve out and 2nd serve in) = $0.59$

(a) Make a tree diagram for the results of the two serves and the outcome (win or lose) of the point.

(b) What is the probability that the serving player wins the point? Show your work.

Short answer

Part (b) $P(Survive5years)=55.8\%$

Part (a) The tree diagram is

Step-by-step solution

Part (a) Step 1. Given Information

In tennis, a player setting has two chances to get a serve into play. Those receiving a new kidney have a $70\%$ chance of living five years, whereas those who return to dialysis have a $50\%$ chance.

Part (a) Step 2. Explanation

There are two choices, therefore at each knot, two branches are needed:

The probability is calculated by dividing the number of favorable outcomes by the total number of possible outcomes:

Part (b) Step 1. Calculation 

Now, for each category of medial risk, determine the probability of paying using a credit card.

$P(1stserve\cap Winpoint)=0.59\times 0.73=0.4307$

$P(1stserveout\cap 2ndserve\cap winpoint)=0.41\times 0.46\times 0.59=0.20$

Fill in the blanks with the corresponding probabilities:

$$\begin{gathered} P\left(winpoint\right)=0.4307+0.20034 \\ =0.63734 \\ =63.734\% \end{gathered}$$