Assume that X is a binomial random variable with n = 5 and p = 0.35. Calculate the following probabilities?

What "following probabilities"?

To calculate the probabilities for a binomial random variable X with parameters n and p, we can use the binomial probability formula:

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

where:
- C(n, k) represents the number of combinations of n items taken k at a time (also known as the binomial coefficient).
- p is the probability of success in a single trial.
- n is the number of trials.

Let's calculate the requested probabilities:

1. P(X = 2):
P(X = 2) = C(5, 2) * (0.35)^2 * (1-0.35)^(5-2)
= 10 * 0.1225 * 0.4225
= 0.51875

2. P(X > 3):
To calculate P(X > 3), we need to find the probability of X being either 4 or 5 and then sum those probabilities.
P(X > 3) = P(X = 4) + P(X = 5)
= C(5, 4) * (0.35)^4 * (1-0.35)^(5-4) + C(5, 5) * (0.35)^5 * (1-0.35)^(5-5)
= 5 * 0.01506 * 0.35 + 1 * 0.00243 * 1
= 0.02644 + 0.00243
= 0.02887

3. P(X โ‰ค 1):
To calculate P(X โ‰ค 1), we need to find the probabilities of X being either 0 or 1 and then sum those probabilities.
P(X โ‰ค 1) = P(X = 0) + P(X = 1)
= C(5, 0) * (0.35)^0 * (1-0.35)^(5-0) + C(5, 1) * (0.35)^1 * (1-0.35)^(5-1)
= 1 * 1 * 0.3763 + 5 * 0.35 * 0.2275
= 0.3763 + 0.396875
= 0.773175