Question

A researcher has determined that an experiment has a success rate of 85%. If the experiment is repeated 6 times, find the probability that:

a) It will be successful at least two times P(x > 2)
b) It will be successful at least five times P(x > 5)

Answer 1

To find the probability of at least two successful outcomes in an experiment with an 85% success rate over 6 trials, use the binomial probability formula. The probability of at least two successes is 0.9997. To find the probability of at least five successes, calculate the probabilities of five and six successes and subtract from 1, resulting in a probability of 0.2979.

Step-by-step explanation:

To find the probability that an experiment with a success rate of 85% will be successful at least two times out of 6, we can use the binomial probability formula. The probability of a success is 0.85, so the probability of a failure is 1 - 0.85 = 0.15. We can calculate the probabilities for 0, 1, and 2 successes and then subtract from 1 to find the probability of at least two successes:

1. P(x = 0) = binomialpdf(6, 0.85, 0) = 0.0003087
2. P(x = 1) = binomialpdf(6, 0.85, 1) = 0.004371
3. P(x = 2) = binomialpdf(6, 0.85, 2) = 0.03263

P(x > 2) = 1 - (P(x = 0) + P(x = 1) + P(x = 2)) = 0.9997183

To find the probability that the experiment will be successful at least five times out of 6, we can calculate the probabilities for 5 and 6 successes and then subtract from 1:

1. P(x = 5) = binomialpdf(6, 0.85, 5) = 0.327079
2. P(x = 6) = binomialpdf(6, 0.85, 6) = 0.3749816

P(x > 5) = 1 - (P(x = 5) + P(x = 6)) = 0.2979394