Anne, bob, and carol- all biostatisticians and epidemiologists- are emailing each other about plans for the...

Anne, bob, and carol- all biostatisticians and epidemiologists- are emailing each other about plans for the weekend. Anne suggests that they all go surfing to Santa Cruz, but bob writes back indicating that he is reluctant to do so because he is terrified of sharks. Carol responds by arguing that P(great weekend | surfing) is 0.80 and P(great weekend | no surfing) is 0.50. Assuming that Carol’s probabilities are correct, further assume for this example that each person’s weekend happiness is independent of any other person’s weekend happiness:

1. Given that all friends go surfing, what is the probability that all 3 of them have a great weekend?

2. Given that all 3 friends go surfing, what is the probability that 2 of them have a great weekend and 1 of them does not have a great weekend? (Hint, how many ways could 2 of the friends have a great weekend?)

Solutions

Expert Solution

The probability that Anne,Bob, and Carol have a great weekend given that they go surfing is 0.80

P(great weekend | surfing) = 0.80

This probability is the same for each of Anne,Bob, and Carol and this probability is independent of others.

1. the conditional probability that all 3 of them have a great weekend, given that they all go surfing is

ans: Given that all friends go surfing, the probability that all 3 of them have a great weekend is 0.512

2. Given that there are 3 friends, number of ways 2 of them can have a great week end is

Given that all 3 friends go surfing, the probability that 2 of them have a great weekend is

ans: Given that all 3 friends go surfing, what is the probability that 2 of them have a great weekend and 1 of them does not have a great weekend is 0.384

We can also solve the above using Binomial distribution.

Let X be the number out of 3 friends (Anne,Bob, Carole) who had a great week end, given that they go surfing.

We can say that X has a Binomial distribution with parameters, number of trials (number of friends) n=3 and success probability (the probability that any one of them had a great weekend given that they go surfing) p=0.80.

The Binomial probability that X=x out of 3 friends had a great week end, given that they go surfing is

1. Given that all friends go surfing, the probability that all 3 of them have a great weekend is the probability that X=3

2. Given that all 3 friends go surfing, what is the probability that 2 of them have a great weekend and 1 of them does not have a great weekend is the probability that X=2

orchestra answered 1 year ago

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