Question

Medical risks Morris’s kidneys are failing, and he is awaiting a kidney transplant. His doctor gives him this information for patients in his condition: 90% survive the transplant and 10% die. The transplant succeeds in 60% of those who survive, and the other 40% must return to kidney dialysis. The proportions who survive five years are 70% for those with a new kidney and

50% for those who return to dialysis.

(a) Make a tree diagram to represent this setting.

(b) Find the probability that Morris will survive for five years. 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

Ninety percent of those who receive a kidney transplant survive, but ten percent die. The transplant is successful in $60\%$of individuals who survive, while the remaining $40\%$ must return to dialysis. Those who receive a new kidney have a $70\%$ chance of surviving 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:

$Theprobability=\frac{Thenumberoffavorableoutcomes}{Thetotalnumberofpossibleoutcomes}$

Part (b) Step 1. Concept Used

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

Part (b) Step 2. Calculation

Now, for each sort of medical risk, determine the likelihood of paying using a credit card. $P(Survive5years\cap Succeeds\cap Survive)=90\%\times 60\%\times 70\%$

$=0.9\times 0.6\times 0.7=0.3780$

$P(Survive5years\cap Returntodialysis\cap Survive)=90\%\times 40\%\times 50\%$

$=0.9\times 0.4\times 0.5=0.1800$

Fill in the blanks with the corresponding probabilities:

$$\begin{gathered} P(Survive5years)=0.3780+0.1800 \\ P(Survive5years)=0.558 \\ P(Survive5years)=55.8\% \end{gathered}$$