You and your friend Anderson both take the bus home after class. Every Monday, W

Question

You and your friend Anderson both take the bus home after class. Every Monday, Wednesday and Friday, you and Anderson will take one of the 5:10 p.m., 5:15 p.m., or 5:20 p.m. buses. You are equally likely to take any of the three buses. Anderson is more likely to take the 5:10 p.m. bus (with probability 1/2) than 5:15 p.m. and 5:20 p.m. bus (with probability 1/4 each). Assume that the bus rides are independent from each other.

a) What is the probability that you and Anderson will have attended no more than 3 lectures before you both take the same bus home for the first time?

b) What is the expected number of lectures you and Anderson will have attended until you both take the same bus home for the third time? Find the variance as well.

Explanation / Answer

probabilty that you both catch the same bus ==P(both catch bus 1+both catch bus 2+both catch bus 3)

=(1/3)(1/2)+(1/3)*(1/4)+(1/3)*(1/4)=1/3

a) hence probabilty it took at most 3 lectures for you both to catch the same bus =P(you catch bus after 1st lecture+not after fist but second+not after first two but third) =(1/3)+(2/3)*(1/3)+(2/3)*(2/3)*(1/3)=0.7037

c) from negative binomial distribution expected number of lectures=r/p=3/(1/3)=9

and variance =r(1-p)/p2 =18

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