Question

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?

   (Enter factorials by typing for example fact(n) for n!. Do not worry if the parser does not display correctly; the grader will work independently. If you wish to have proper display, enclose any factorial by parentheses, e.g. (fact(n)).)

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?

   Yes

   No

Answer 1

The number of balls in the 1st bin is binomially distributed with .

The PMF of is .

The probability,

Each of bins has probability of being empty. The expected number of empty bins is

The possible ways of picking red ball in the 3rd round are

The probability,

Thus, the probability that the ball picked in the third round is red is

We have

We can see that . The events are not independent.