What is the probability that in a sample of the 6 most recent recommendations, that at least 5 of them made money?

Question

An analyst claims that there is a 70% chance of making money on the investments he recommends. If this is true, what is the probability that in a sample of the 6 most recent recommendations, that at least 5 of them made money?

Answer 1

We want the probability that the number of cases where money is made is 5, 6, or 7. Let the number of successes be k. There are 7!/(k! * (7-k)!) ways in which to select k cases from the 7. Each of these cases has probability 0.7^k * 0.3^(7-k). Summing this value for k equal to 5, 6, or 7 gives the answer, approximately 0.6471. To interpret this answer, note that this is a somewhat moderate probability, i.e., not to small and not too large, which means it will be difficult to test whether the investment advisor is correct about his 70% claim, since if an event with probability 0.6471 actually happens, we wouldn't be very surprised. Also, if the event doesn't happen (at least 5 investments don't make money), we also wouldn't be very surprised. So we'd probably have to test this advisor many times!