A recent study has shown that a new treatment cures a certain disease 68% of the time. If this percentage is correct, what is the probability that, in a random sample of 7 patients undergoing this treatment, exactly 4 are cured?Round your answer to three decimal places.
Question
A recent study has shown that a new treatment cures a certain disease 68% of the time. If this percentage is correct, what is the probability that, in a random sample of 7 patients undergoing this treatment, exactly 4 are cured?Round your answer to three decimal places.
Solution
This problem can be solved using the binomial probability formula, which is:
P(X=k) = C(n, k) * (p^k) * ((1-p)^(n-k))
where:
- P(X=k) is the probability of k successes in n trials
- C(n, k) is the combination of n items taken k at a time
- p is the probability of success
- n is the number of trials
- k is the number of successes
In this case:
- n = 7 (the number of patients)
- k = 4 (the number of patients we want to be cured)
- p = 0.68 (the probability of a patient being cured)
First, calculate C(n, k), which is the number of combinations of n items taken k at a time. This can be calculated as:
C(n, k) = n! / [k!(n-k)!]
where "!" denotes factorial, which is the product of all positive integers up to that number. So:
C(7, 4) = 7! / [4!(7-4)!] = 35
Next, calculate p^k and (1-p)^(n-k):
p^k = 0.68^4 = 0.2139 (1-p)^(n-k) = (1-0.68)^(7-4) = 0.032
Finally, substitute these values back into the binomial probability formula:
P(X=4) = C(7, 4) * (0.68^4) * ((1-0.68)^(7-4)) = 35 * 0.2139 * 0.032 = 0.239
So, the probability that exactly 4 out of 7 patients are cured is approximately 0.239, or 23.9%, when rounded to three decimal places.
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