Example 19.5

BSIP/Phototake

EXAMPLE 5 A kidney transplant

Morris’s kidneys have failed and he is awaiting a kidney transplant. His doctor gives him this information for patients in his condition: 90% survive the transplant operation, 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 for at least five years are 70% for those with a new kidney and 50% for those who return to dialysis. Morris wants to know the probability that he will survive for at least five years.

Step 1. The tree diagram in Figure 19.1 organizes this information to give a probability model in graphical form. The tree shows the three stages and the possible outcomes and probabilities at each stage. Each path through the tree leads to either survival for five years or to death in less than five years. To simulate Morris’s fate, we must simulate each of the three stages. The probabilities at Stage 3 depend on the outcome of Stage 2.

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Figure 19.1: Figure 19.1 A tree diagram for the probability model, Examples 5. Each branch point starts a new stage, with outcomes and their probabilities written on the branches. One repetition of the model follows the tree to one of its endpoints.

Step 2. Here is our assignment of digits to outcomes:

Stage 1:

0 = die

1, 2, 3, 4, 5, 6, 7, 8, 9 = survive

Stage 2:

0, 1, 2, 3, 4, 5 = transplant succeeds

6, 7, 8, 9 = return to dialysis

Stage 3 with kidney:

0, 1, 2, 3, 4, 5, 6 = survive five years

7, 8, 9 = die

Stage 3 with dialysis:

0, 1, 2, 3, 4 = survive five years

5, 6, 7, 8, 9 = die

The assignment of digits at Stage 3 depends on the outcome of Stage 2. That’s lack of independence.

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Step 3. Here are simulations of several repetitions, each arranged vertically. We used random digits from line 110 of Table A.

	Repetition 1	Repetition 2	Repetition 3	Repetition 4
Stage 1	3 Survive	4 Survive	8 Survive	9 Survive
Stage 2	8 Dialysis	8 Dialysis	7 Dialysis	1 Kidney
Stage 3	4 Survive	4 Survive	8 Die	8 Die

Morris survives five years in two of our four repetitions. Now that we understand how to arrange the simulation, we should turn it over to a computer to do many repetitions. From a long simulation or from mathematics, we find that Morris has probability 0.558 of living for at least five years.