Three drugs are being tested for use as the treatment of a certain disease. Let p1, p2, and p3 represent the probabilities of success for the respective drugs. As three patients come in, each is given one of the drugs in a random order. After n = 10 “triples” and assuming independence, compute the probability that the maximum number of successes with one of the drugs exceeds eight if, in fact, p1 = p2 = p3 = 0.7.
To compute the probability that the maximum number of successes with one of the drugs exceeds eight, we need to calculate the probability of each possible outcome. Let's break down the steps:
Step 1: Define the probability of success for each drug.
Given that p1 = p2 = p3 = 0.7, we know that the probability of success for each drug is 0.7.
Step 2: Determine the number of possible outcomes.
Since there are three drugs and three patients, there are 3^3 = 27 possible outcomes. Each outcome represents a different order in which the drugs are given to the patients.
Step 3: Identify the outcomes where the maximum number of successes exceeds eight.
In this case, we need to find all the outcomes where the maximum number of successes with one of the drugs (p1, p2, or p3) is greater than eight.
Step 4: Calculate the probability of each outcome.
To calculate the probability of each outcome, we need to multiply the probability of success for each drug together. Since the probabilities are the same for all drugs (0.7), we can simply raise this value to the power of the number of successful outcomes.
Step 5: Calculate the total probability.
To find the probability that the maximum number of successes with one of the drugs exceeds eight, we need to sum up the probabilities of all the outcomes where this condition is met.
Using these steps, we can calculate the probability.