To find the probability of fewer than 3 successes in a binomial experiment, we need to calculate the sum of the probabilities of 0, 1, and 2 successes.

The formula for the probability of k successes in a binomial experiment is:

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

where:
- C(n, k) is the number of combinations of n items taken k at a time,
- p is the probability of success,
- n is the number of trials,
- k is the number of successes.

Let's calculate the probabilities for 0, 1, and 2 successes:

1. For 0 successes (k=0):

P(X=0) = C(13, 0) * (0.4^0) * ((1-0.4)^(13-0))

2. For 1 success (k=1):

P(X=1) = C(13, 1) * (0.4^1) * ((1-0.4)^(13-1))

3. For 2 successes (k=2):

P(X=2) = C(13, 2) * (0.4^2) * ((1-0.4)^(13-2))

Finally, add these probabilities together to get the probability of fewer than 3 successes:

P(X

Question

To find the probability of fewer than 3 successes in a binomial experiment, we need to calculate the sum of the probabilities of 0, 1, and 2 successes.

The formula for the probability of k successes in a binomial experiment is:

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

where:
- C(n, k) is the number of combinations of n items taken k at a time,
- p is the probability of success,
- n is the number of trials,
- k is the number of successes.

Let's calculate the probabilities for 0, 1, and 2 successes:

1. For 0 successes (k=0):

P(X=0) = C(13, 0) * (0.4^0) * ((1-0.4)^(13-0))

2. For 1 success (k=1):

P(X=1) = C(13, 1) * (0.4^1) * ((1-0.4)^(13-1))

3. For 2 successes (k=2):

P(X=2) = C(13, 2) * (0.4^2) * ((1-0.4)^(13-2))

Finally, add these probabilities together to get the probability of fewer than 3 successes:

P(X<3) = P(X=0) + P(X=1) + P(X=2)

This will give you the final answer. Remember to round to at least three decimal places.

Knowee AI · Accepted Answer