Question

A Bernoulli Trials experiment consists of 4 trials, with a 1/3 probability of success on each trial.

What is the probability of at least 1 success given there are failures?

What is the probability there are failures given there are successes?

Show Work

This is the question as it was asked in class.

What is the probability that there is at least 1 success given that there are any failures?

What is the probability that there are any failures given that there are any successes?

Answer 1

1)

P(there are failures) =1-P(all success) =1-(1/3)⁴
P(there are failures and success)= 1-P(all success)-P(all failures) =1-(1/3)⁴-(2/3)⁴

therefore P( at least 1 success given there are failures )

=P(there are failures and success)/P(there are failures)

=(1-(1/3)⁴-(2/3)⁴ )/(1-(1/3)⁴ )

=0.80

2)

P(there are success) =1-P(all failures) =1-(2/3)⁴
P(there are failures and success)= 1-P(all success)-P(all failures) =1-(1/3)⁴-(2/3)⁴

therefore P( at least 1 failure given there are success )

=P(there are failures and success)/P(there are Success)

=(1-(1/3)⁴-(2/3)⁴)/(1-(2/3)⁴)

=0.9846