Question

A new prescription sleep-aid drug used to treat insomnia claims that 85% of the patients who take the drug experience no side effects at all. In a small study of 20 patients, what is the probability that at least 85% of the patients will experience no side effects?

Answer 1

To find the probability that at least 85% of the patients will experience no side effects, we use the binomial distribution. We calculate the cumulative probability using the formula P(X ≥ k) = 1 - P(X < k), where X is the number of patients experiencing no side effects and k is the minimum number of patients we want to have no side effects.

Step-by-step explanation:

To find the probability that at least 85% of the patients will experience no side effects, we need to use the binomial distribution. In this case, we have a sample size of 20 patients and a success rate of 85% (0.85) for no side effects. To calculate the probability, we can use the cumulative binomial probability formula:

P(X ≥ k) = 1 - P(X < k)

where X is the number of patients experiencing no side effects and k is the minimum number of patients we want to have no side effects.

Let's calculate the probability:

1. P(X < k) = P(X = 0) + P(X = 1) + ... + P(X = k-1)
2. P(X ≥ k) = 1 - P(X < k)

Plugging in the values, we have:

P(X < k) = P(X = 0) + P(X = 1) + ... + P(X = k-1) = (0.15)^0 * (0.85)^20 + (0.15)^1 * (0.85)^19 + ... + (0.15)^(k-1) * (0.85)^(20-k+1)

P(X ≥ k) = 1 - P(X < k)

Now, we can calculate the probability using this formula.