Three drugs are being tested for use as the treatment of a certain disease.
Let p 1 , p 2 , and p 3 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.
Three drugs are being tested for use as the treatment of a certain disease.
Let p 1 , p 2 , and p 3 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 can use the binomial distribution.
The probability of success for each drug is given as p1 = p2 = p3 = 0.7.
Let X be the random variable representing the maximum number of successes with one of the drugs. We need to find P(X > 8).
Since there are three drugs and each patient is given one of the drugs randomly, the probability distribution of X can be modeled as a negative hypergeometric distribution.
P(X > 8) = 1 - P(X <= 8)
To find P(X <= 8), we need to calculate the probability of getting 0, 1, 2, ..., 8 successes with one of the drugs and then sum them up.
P(X <= 8) = P(X = 0) + P(X = 1) + P(X = 2) + ... + P(X = 8)
Using the binomial distribution formula, where n is the number of trials and k is the number of successes, the probability of getting k successes out of n trials is given by:
P(X = k) = C(n, k) * (p ^ k) * ((1 - p) ^ (n - k))
Where C(n, k) is the binomial coefficient, given by n! / (k! * (n - k)!)
In this case, n = 10 (number of "triples") and p = 0.7 (probability of success for each drug).
Now we can calculate P(X = 0), P(X = 1), P(X = 2), ..., P(X = 8) using the formula and sum them up to find P(X <= 8). Finally, subtracting this probability from 1 will give us the desired result, P(X > 8).
Let's calculate it step by step:
Step 1: Calculate P(X = 0)
P(X = 0) = C(10, 0) * (0.7 ^ 0) * ((1 - 0.7) ^ (10 - 0))
= 1 * 1 * (0.3 ^ 10)
= 0.3 ^ 10
Step 2: Calculate P(X = 1)
P(X = 1) = C(10, 1) * (0.7 ^ 1) * ((1 - 0.7) ^ (10 - 1))
= 10 * 0.7 * (0.3 ^ 9)
Step 3: Calculate P(X = 2)
P(X = 2) = C(10, 2) * (0.7 ^ 2) * ((1 - 0.7) ^ (10 - 2))
= 45 * (0.7 ^ 2) * (0.3 ^ 8)
Continue this process until P(X = 8) and sum up all the probabilities.
Finally, subtract the result from 1 to find P(X > 8).
To solve this problem, we need to calculate the probability of the maximum number of successes with one of the drugs exceeding eight out of ten "triples" given that p1 = p2 = p3 = 0.7.
Let's break down the problem into steps:
Step 1: Find the probability of success for each drug.
Since p1 = p2 = p3 = 0.7, the probability of success for any drug is 0.7.
Step 2: Determine the probability of getting the maximum number of successes with one drug exceeding eight out of ten "triples."
To do this, we need to calculate the probability of each possible outcome: 9, 10, and 0-8 successes with each drug. Then, we sum up the probabilities of the outcomes where the maximum number of successes with one drug exceeds eight.
For each possible outcome, the probability can be calculated using the binomial distribution formula:
P(X = k) = C(n, k) * p^k * (1-p)^(n-k),
where:
- P(X = k) is the probability of getting exactly k successes in n trials.
- C(n, k) is the binomial coefficient, which represents the number of ways to choose k successes out of n trials.
- p is the probability of success for each trial.
- n is the number of trials.
In this case, since we want to find the probability of the maximum number of successes exceeding eight, we sum the probabilities for k = 9 and k = 10, as follows:
P(max > 8) = P(X = 9) + P(X = 10)
P(X = 9) = C(10, 9) * (0.7)^9 * (1-0.7)^(10-9)
P(X = 10) = C(10, 10) * (0.7)^10 * (1-0.7)^(10-10)
Step 3: Calculate the values for P(X = 9) and P(X = 10).
Using the binomial coefficient and probability calculations:
P(X = 9) = 10 * (0.7)^9 * (0.3)^1
P(X = 10) = 1 * (0.7)^10 * (0.3)^0
Step 4: Sum up the probabilities to get the final result.
P(max > 8) = P(X = 9) + P(X = 10)
Now, you can compute the probabilities using the values obtained in the previous steps to find the desired probability.