Chapter 5.3, Problem 87E

a.

To determine

To Draw: a tree diagram to represent the given setting.

Explanation of Solution

Given:

The given condition is 90% survive the kidney transplant but 10% die.

The transplant succeeds in 60% of those who survive, and other 40% must return to kidney dialysis.

The proportion who survive five years are 70% for those with a new kidney and 50% for those who return to dialysis.

Calculation:

There are two options, thus two branches are nedded at each knot:

The probability is the number of favourable outcomesdivided by the number of possible outcomes:

  

b.

To determine

To find: the probability that Morris will survive for 5 years.

Answer to Problem 87E

  $P\left(\text{Survive 5 years}\right)=55.8\%$

Explanation of Solution

Given:

The given condition is 90% survive the kidney transplant but 10% die.

The transplant succeeds in 60% of those who survive, and other 40% must return to kidney dialysis.

The proportion who survive five years are 70% for those with a new kidney and 50% for those who return to dialysis.

Concept used:

Multiplication rule $P\left(A\cap B\cap C\right)=P\left(A\right)\times P\left(B\right)\times P\left(C\right)$

Calculation:

Now, to find the probability of paying with credit card for each type of medial risk.

  $\begin{array}{l}P\left(\text{Survive 5 years}\cap \text{Succeeds}\cap \text{Survive}\right)=90\%\times 60\%\times 70\%\\ =0.9\times 0.6\times 0.7=0.3780\end{array}$

  $\begin{array}{l}P\left(\text{Survive 5 years}\cap \text{Return to dialysis}\cap \text{Survive}\right)=90\%\times 40\%\times 50\%\\ =0.9\times 0.4\times 0.5=0.1800\end{array}$

Add the corresponding probabilities:

  $\begin{array}{l}P\left(\text{Survive 5 years}\right)\\ =0.3780+0.1800\\ =0.558=55.8\%\end{array}$ .