Morgan Stanley - Nov 17, 2008

[Phone interview]

1. There are two children in a family and one of them is a boy. What's the probability that the other one is a girl?

Sol: A sample space for the event of a family with two children could be the set of touples {(B,B), (B,B), (G,B), (G,G)}. We are given that one of them is a boy, hence the conditional sample space would be {(B,B), (B,B), (G,B)} and thus the probability that the other child is a girl is 2/3.

2. What's the expected minimum number of tosses you need to get three consecutive heads in a row?

Sol: Tricky. A rigorous solution would be pretty long (I attempted it and couldn't finish it in the time given), but after the interview I found an easier one. Here it is. Let N be the expected minimum number of tosses you need to get three consecutive heads in a row and condition on the first three outcomes as follows:

1. If you get a T (tail), then the expected minimum number of tosses you need to get three consecutive heads in a row increases by one. The probability of having T is 1/2.
2. If you get a H (head) and then a T, then the expected minimum number of tosses you need to get three consecutive heads in a row increases by two. The probability of having HT is 1/4.
3. If you get HH in the first two toses and T in your third one, then the expected minimum number of tosses you need to get three consecutive heads in a row increases by three. The probability of having HHT is 1/8.
4. But if you get HHH, then you only took three tosses to get three consecutive heads in a row. The probability of having HHH is 1/8.

Thus,

N = (1/2)(N+1) + (1/4)(N+2) + (1/8)(N+3) + (1/8)(3)

which implies that N=14. (This solution assumes N is not infinity, and that's why a more rigorous solution is needed, although I am sure they are not looking for that on the phone)

3. There is a party and everybody shakes hands with everybody. There were 66 hand shakes. How many people were at the party?

Sol: If there are N people, let's line them up. Person 1 hand shakes with N-1 people to his right, Person 2 hand shakes with N-2 people to his right, and so on. Thus there are a total number of

(N-1) + (N-2) + (N-3) + ... + 2 + 1

hand shakes. There are N-1 terms above. Thus in total there are N(N-1)/2 hand shakes. If that number is 66, then N has to be 12.

4. It takes 1.5 people to drink 1 liter of vodka in 1.5 hours. How many people would be needed to drink 0.4 liters in 45 minutes?

Sol: The question, assuming constant rates and people drink at the same rate (etc, etc, etc), is the same as finding out how many people would be needed to drink 0.8 liters in 1.5 hours. Then the time information is not important anymore. So the question now is how many people would be needed to drink 0.8 liters given that 1.5 people drink 1 liter. A straightforward calculation says 1.2 people are needed. (I got very confused with this stupid question... it was so easy. I blew it)