albino rats used to study the hormonal regulation of a metabolic pathway are injected with a drug that inhibits body synthesis of protein. usually, 4 out of 20 rats die from the drug before the experiment is over. this time round, 10 animals are treated with the drug. what is the probability that 5 of these animals will die before the experiment is over?

0.0264

0.2 exactly 5, or at least 5?

To find the probability that 5 of the 10 animals treated with the drug will die before the experiment is over, we can use the binomial probability formula. The formula for binomial probability is:

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

Where:
- P(X = k) is the probability of exactly k successes
- n is the total number of trials
- k is the number of desired successes
- p is the probability of success in a single trial
- (n choose k) is the binomial coefficient, calculated as n! / (k! * (n - k)!)

In this case, we want to find the probability that 5 animals will die (k = 5) out of a total of 10 animals (n = 10). The probability of an animal dying from the drug is given as 4 out of 20 (p = 4/20 = 0.2).

Using the formula:

P(X = 5) = (10 choose 5) * (0.2)^5 * (1 - 0.2)^(10 - 5)

Let's calculate this:

P(X = 5) = (10! / (5! * (10 - 5)!)) * (0.2)^5 * (1 - 0.2)^5

Simplifying:

P(X = 5) = (252) * (0.2)^5 * (0.8)^5

P(X = 5) ≈ 0.026

Therefore, the probability that exactly 5 animals out of the 10 treated with the drug will die before the experiment is over is approximately 0.026, or 2.6%.