# Probability

1. We have a bag that contains n red balls and n blue balls. At each of 2n rounds we remove one of the balls from the bag randomly, and place it in one of available n bins. At each round, each one of the balls that remain in the bag is equally likely to be picked, as is each of the bins, independent of the results of previous rounds. Let Nk be the number of balls in the k -th bin after 2n rounds, i.e., after all balls have been placed in the bins.

1. Find the probability that N1=0 , i.e., that the first bin is empty after all balls have been removed and placed into bins.

2. What is the PMF pN1(k) of N1 ?

3. What is the expected number of empty bins?

4. What is the probability that the ball picked in the third round is red?

5. Let Ri denote the event that i -th ball picked is red. Are the events R1 and R2 independent?

1. 👍 4
2. 👎 0
3. 👁 2,413
1. 1. ((n-1)/n)^n
2. k!/(n-k)!*p^n*(1-p)^n-k
3. n*((n-1)/n)^n
4.
5. No

1. 👍 5
2. 👎 20
2. 1) ((n-1)/n)^n
2) (n!/(n-k)!*k!)(1/n)^n(1-1/n)^n
3) n*((n-1)/n)^n
4) 4/5 = 0.8
5) No

1. 👍 0
2. 👎 12
3. Isnt ans for 1st and 3rd question have the power 2n since we have (n+n) balls and 2n draws for the same

1. 👍 9
2. 👎 0
4. 3 = 1/2

1. 👍 1
2. 👎 1
5. Cant get the 4/5=0.8 on the 4th. How do you get to this

1. 👍 1
2. 👎 0
6. On 1st and 3rd I have 2n as power as well. There are 2n rounds.

1. 👍 1
2. 👎 0
7. on 2 it should be 2n pick k. then imo

2) (2n!/(2n-k)!*k!)(1/n)^2n(1-1/n)^2n-k

1. 👍 1
2. 👎 2
8. Also found 1/2 for 3.

1. 👍 0
2. 👎 1

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