Question

A medical research team wishes to evaluate two different treatments for a disease. Subjects are selected two at a time, and then one of the pair is assigned to each of the two treatments. The treatments are applied, and each is either a success (S) or a failure (F). The researchers keep track of the total number of successes for each treatment. They plan to continue the chance experiment until the number of successes for one treatment exceeds the number of successes for the other treatment by \(2 .\) For example, they might observe the results in the table below. The chance experiment would stop after the sixth pair, because Treatment 1 has 2 more successes than Treatment \(2 .\) The researchers would conclude that Treatment 1 is preferable to Treatment \(2 .\) Suppose that Treatment 1 has a success rate of \(.7\) (i.e., \(P(\) success \()=.7\) for Treatment 1 ) and that Treatment 2 has a success rate of \(.4\). Use simulation to estimate the probabilities in Parts (a) and (b). (Hint: Use a pair of random digits to simulate one pair of subjects. Let the first digit represent Treatment 1 and use \(1-7\) as an indication of a

Short answer

It's not possible to provide a numerical short answer for this question because it depends on the simulation outcome. It must be noted that the expected probability for Treatment 1 is higher than Treatment 2. This might lean towards a conclusion where Treatment 1 stops first, with 2 more successes than Treatment 2. The exact numerical probabilities would depend on the simulated experiments.

Step-by-step solution

Understanding the problem

The purpose of this exercise is to determine probabilities of success for two different treatments, Treatment 1 and Treatment 2. Their success rates are given as .7 and .4 respectively. The result interpretation is to be made when the number of successes of one treatment exceeds the other's by 2.

Simulating results for Treatment 1

Let's simulate a large number of experiments (say, 10,000) for Treatment 1 using random numbers. The digits between 1 and 7 will be considered a 'success', simulating the .7 success rate or 70% probability of Treatment 1. Count the number of successes.

Simulating results for Treatment 2

Execute the same process for Treatment 2. This time, consider the digits between 1 and 4 as 'success', which would simulate the .4 success rate or 40% probability of Treatment 2. Keep track of the number of successes.

Comparison and Concluding Results

Compare the amount of successes of both treatments. The experiment ends when the number of successes for one treatment exceed the other by 2. Analyze the simulation results and compute the probabilities. This can be given by the count of experiments where one treatment exceeds the other by 2 successes over the total count of simulated experiments.