A car rental company has a fleet of 200 cars. They find that on average, 15% of their cars are rented out each day. What is the probability that exactly 35 cars will be rented out on a given day?
To solve this problem, we will use the binomial probability formula, which calculates the probability of obtaining a specific number of successes in a fixed number of trials with a constant probability of success for each trial.

In this case, our "success" is renting out a car, and our "trial" is selecting one of the 200 cars. We want to find the probability of renting out exactly 35 cars.

The binomial probability formula is:

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

where:
- P(X=k) is the probability of obtaining k successes in n trials
- C(n,k) is the number of combinations of choosing k items from a set of n items
- p is the probability of success for each trial
- n is the total number of trials
- k is the desired number of successes

For our problem, we have:

n = 200 (total number of cars)
k = 35 (desired number of rented cars)
p = 0.15 (probability of renting out a car)

First, let's calculate C(200,35), which is the number of ways to choose 35 cars from a set of 200:

C(200,35) = 200! / (35!(200-35)!)
               ≈ 3.162 * 10^42

Now, we can plug these values into the binomial probability formula:

P(X=35) = C(200,35) * p^35 * (1-p)^(200-35)
              = 3.162 * 10^42 * (0.15)^35 * (0.85)^165
              ≈ 0.0487

So, the probability that exactly 35 cars will be rented out on a given day is approximately 0.0487 or 4.87%.