If a treatment is effective in 80% of cases, what is the probability that it wil

If a treatment is effective in 80% of cases, what is the probability that it will be effective for 3 out of 5 patients? (2020)

If a treatment is effective in 80% of cases, what is the probability that it will be effective for 3 out of 5 patients? (2020)

Step 1: Identify the total number of patients (n). In this case, n = 5.
Step 2: Identify the number of patients for whom the treatment is effective (k). Here, k = 3.
Step 3: Identify the probability that the treatment is effective for one patient (p). In this case, p = 0.8.
Step 4: Calculate the probability that the treatment is not effective for one patient (1 - p). This is 1 - 0.8 = 0.2.
Step 5: Use the binomial probability formula: P(X=k) = C(n,k) * p^k * (1-p)^(n-k).
Step 6: Calculate C(n, k), which is the number of combinations of n items taken k at a time. C(5, 3) = 5! / (3! * (5-3)!) = 10.
Step 7: Calculate p^k, which is (0.8)^3 = 0.512.
Step 8: Calculate (1-p)^(n-k), which is (0.2)^2 = 0.04.
Step 9: Multiply the results: P(3 out of 5) = C(5, 3) * (0.8)^3 * (0.2)^2 = 10 * 0.512 * 0.04.
Step 10: Calculate the final result: 10 * 0.512 * 0.04 = 0.2048.

Binomial Probability – The question tests the understanding of the binomial probability formula, which calculates the likelihood of a certain number of successes in a fixed number of independent trials.