A gambler decides to keep betting on red at roulette, and stop as soon as she has won a total of five bets.
a) what is the probability that she has to make exactly 8 bets before stopping?
b) what is the probability that she has to make at least 9 bets?
Then call the value p. You can chase up the value from your instructor.
Now use the negative binomial distribution: Negative binomial distribution - Wikipedia, the free encyclopedia
I'll do it using only the binomial distribution (you might not be familiar with the negative binomial distribution).
a) You need exactly 2 successes in the first 7 trials and then a succes on the 8th trial.
Let X ~ Binomial(n = 7, p = 1/2).
Then Pr(8 trials) = .
b) Pr(9 or more trials) = 1 - Pr(8 or less trials) = 1 - Pr(8 trials) - Pr(7 trials) - ..... - Pr(4 trials) - Pr(3 trials).
Calculate Pr(7 trials), Pr(6 trials), ...... Pr(3 trials) in a similar way to how Pr(8 trials) was calculated.