To solve this problem, we will use the binomial probability formula:

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

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

We want to find the probability of 1 or fewer successes, so we need to calculate P(X=0) and P(X=1), then add these two probabilities together.

1. Calculate P(X=0):
- C(n, k) = C(11, 0) = 1 (since the combination of n items taken 0 at a time is always 1)
- p^k = 0.32^0 = 1 (since any number to the power of 0 is 1)
- (1-p)^(n-k) = (1-0.32)^(11-0) = 0.68^11 ≈ 0.00057
- So, P(X=0) = 1 * 1 * 0.00057 = 0.00057

2. Calculate P(X=1):
- C(n, k) = C(11, 1) = 11 (since the combination of 11 items taken 1 at a time is 11)
- p^k = 0.32^1 = 0.32
- (1-p)^(n-k) = (1-0.32)^(11-1) = 0.68^10 ≈ 0.00084
- So, P(X=1) = 11 * 0.32 * 0.00084 = 0.00296

Finally, add P(X=0) and P(X=1) together to get the probability of 1 or fewer successes:

P(X ≤ 1) = P(X=0) + P(X=1) = 0.00057 + 0.00296 = 0.00353

So, the probability that the experiment results in 1 or fewer successes is approximately 0.00353, or 0.353%.

Question

To solve this problem, we will use the binomial probability formula:

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

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

We want to find the probability of 1 or fewer successes, so we need to calculate P(X=0) and P(X=1), then add these two probabilities together.

1. Calculate P(X=0):
   - C(n, k) = C(11, 0) = 1 (since the combination of n items taken 0 at a time is always 1)
   - p^k = 0.32^0 = 1 (since any number to the power of 0 is 1)
   - (1-p)^(n-k) = (1-0.32)^(11-0) = 0.68^11 ≈ 0.00057
   - So, P(X=0) = 1 * 1 * 0.00057 = 0.00057

2. Calculate P(X=1):
   - C(n, k) = C(11, 1) = 11 (since the combination of 11 items taken 1 at a time is 11)
   - p^k = 0.32^1 = 0.32
   - (1-p)^(n-k) = (1-0.32)^(11-1) = 0.68^10 ≈ 0.00084
   - So, P(X=1) = 11 * 0.32 * 0.00084 = 0.00296

Finally, add P(X=0) and P(X=1) together to get the probability of 1 or fewer successes:

P(X ≤ 1) = P(X=0) + P(X=1) = 0.00057 + 0.00296 = 0.00353

So, the probability that the experiment results in 1 or fewer successes is approximately 0.00353, or 0.353%.

Knowee AI · Accepted Answer

To solve this problem, we will use the binomial probability formula:

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

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

We want to find the probability of 1 or fewer successes, so we need to calculate P(X=0) and P(X=1), then add these two probabilities together.

1. Calculate P(X=0):
   - C(n, k) = C(11, 0) = 1 (since the combination of n items taken 0 at a time is always 1)
   - p^k = 0.32^0 = 1 (since any number to the power of 0 is 1)
   - (1-p)^(n-k) = (1-0.32)^(11-0) = 0.68^11 ≈ 0.00057
   - So, P(X=0) = 1 * 1 * 0.00057 = 0.00057

2. Calculate P(X=1):
   - C(n, k) = C(11, 1) = 11 (since the combination of 11 items taken 1 at a time is 11)
   - p^k = 0.32^1 = 0.32
   - (1-p)^(n-k) = (1-0.32)^(11-1) = 0.68^10 ≈ 0.00084
   - So, P(X=1) = 11 * 0.32 * 0.00084 = 0.00296

Finally, add P(X=0) and P(X=1) together to get the probability of 1 or fewer successes:

P(X ≤ 1) = P(X=0) + P(X=1) = 0.00057 + 0.00296 = 0.00353

So, the probability that the experiment results in 1 or fewer successes is approximately 0.00353, or 0.353%.

A binomial experiment with probability of success p=0.32 and n=11 trials is conducted. What is the probability that the experiment results in 1 or fewer successes?2 points

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