help me to provide the step by step complete solution of the problem, formulas used and their values at each step, explain it like you have to explain it to a child: A new drug is in the process of being approved by the FDA. The drug is being tested in 15 mice, and there is a 8% chance that a mice will die from the drug in the trial. The drug will be approved for the next stage if four or less mice die in the trial. Calculate the probability the drug will be approved.

Sure, let's break this down into simple steps. We're going to use a formula called the binomial probability formula. This formula is used when you have a certain number of "trials" (in this case, the 15 mice), and you want to know the probability of a certain number of "successes" (in this case, a mouse dying is considered a "success", even though it's not a good thing).

The binomial probability formula is:

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

where:

• P(X=k) is the probability we want to find.
• C(n, k) is the number of combinations of n items taken k at a time.
• p is the probability of success on a single trial.
• n is the number of trials.
• k is the number of successes we're interested in.

Let's plug in the numbers:

• n = 15 (the number of mice)
• p = 0.08 (the chance a mouse will die)
• k = 0, 1, 2, 3, 4 (we want to know the probability that four or less mice die)

We need to calculate the probability for each value of k from 0 to 4, and then add those probabilities together.

Let's start with k=0:

P(X=0) = C(15, 0) * (0.08^0) * ((1-0.08)^(15-0))

C(15, 0) is the number of ways to choose 0 mice out of 15, which is 1. 0.08^0 is 1, because any number to the power of 0 is 1. (1-0.08)^(15-0) is (0.92)^15, which is about 0.196.

So, P(X=0) = 1 * 1 * 0.196 = 0.196.

We repeat this process for k=1, k=2, k=3, and k=4. Then we add all those probabilities together to get the total probability that four or less mice die.

This is a bit of a long process, but it's the correct way to solve this problem. If you have a calculator that can do binomial probability calculations, it can save you a lot of time.