A generic drug used to treat a particular condition has a success rate of 62%. A random sample of 12 people with this particular condition are treated with the drug. Calculate the probability that  (i)  Exactly 6 people are treated successfully (Give the answer correct to 3 decimal places)

To solve this problem, we can use 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,
• 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, we have:

• n = 12 (the number of people treated),
• k = 6 (the number of people treated successfully),
• p = 0.62 (the success rate of the drug).

Substituting these values into the formula, we get:

P(X=6) = C(12, 6) * (0.62^6) * ((1-0.62)^(12-6))

First, calculate C(12, 6), which is the number of combinations of 12 items taken 6 at a time. This can be calculated as:

C(12, 6) = 12! / (6!(12-6)!) = 924

Next, calculate (0.62^6) = 0.0276 and ((1-0.62)^6) = 0.0014.

Finally, substitute these values back into the formula to get:

P(X=6) = 924 * 0.0276 * 0.0014 = 0.035

So, the probability that exactly 6 out of 12 people are treated successfully with the drug is approximately 0.035 or 3.5% when rounded to three decimal places.

This is a binomial distribution problem. The formula for binomial distribution is:

P(X=k) = C(n, k) * (p^k) * ((1-p)^(n-k))

where:

• P(X=k) is the probability of k successes,
• 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.

Given:

• n = 12 (number of people treated with the drug),
• p = 0.62 (probability of success),
• k = 6 (number of successful treatments).

We can substitute these values into the formula:

P(X=6) = C(12, 6) * (0.62^6) * ((1-0.62)^(12-6))

First, calculate C(12, 6), which is the number of combinations of 12 items taken 6 at a time. This can be calculated as:

C(12, 6) = 12! / (6!(12-6)!) = 924

Next, calculate (0.62^6) = 0.0276 and ((1-0.62)^(12-6)) = 0.0085.

Finally, substitute these values back into the formula:

P(X=6) = 924 * 0.0276 * 0.0085 = 0.216 (rounded to three decimal places).

So, the probability that exactly 6 out of 12 people are treated successfully with the drug is 0.216.