A roulette wheel in the U.S. contains 38 equally sized spaces. Two spaces are gr

Question

A roulette wheel in the U.S. contains 38 equally sized spaces. Two spaces are green, 18 are red and the other 18 are black. a). You are going to play roulette for nine rounds and you are going to bet on red in each round. i). What is the probability that you win in exactly five of those rounds? (3 points) ii). What is the probability that you win in at least seven of those rounds? (3 points) iii). What is the probability that you do not win at all? (3 points) iv). What is the expected number of wins that you should get? (2 points) b). Now, you are going to play roulette for as many rounds as possible until the roulette hits red for the first time. i). What is the probability that you get the first red on the fifth round? (3 points) ii). What is the expected number of rounds that you would need to play until you get a red? (3 points) a red? (8 points) (3 points) iii). What is the probability that you have to play for more than three rounds to get iv). What is the probability that you get the first red on the second round or before?

Explanation / Answer

b)

this is geometric distribution with paramter p =probability of red =18/38 =9/19

(i) probability to get first red on fifth round =P(first 4 round no red and on fifth rounf first red)

=(10/19)4*(9/19)=0.0363

(ii) expected number =1/p =1/(9/19) =2.1111

(iii) P(X>3) =P(till 3 round no red) =(10/19)3 =0.1458

(iv)P(X<=2) =1-P(till 2nd round no red) =1-(10/19|)2 =1-0.2770 =0.7230

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