Question

2. A jar contains 5 balls, 4 of which are blue and one red.

(a) If you draw balls one at a time and replace them, what is the expected draw at which

you will see the red ball?

(b) If you draw balls one at a time, but without replacing them, what is the expected time

to see the red ball?

3. A jar contains 1 red ball and an unknown number of blue balls. You make 20 draws with

replacement from the jar. What is a maximum likelihood estimator of the number of blue

balls in the jar?

Answer 1

2.

(a)

Suppose, random variable X denotes number of draws required to get red ball.

As we draw with replacement, X can take any value from 1 to infinity.

Now, probability to be selected in i th draw

[ Same as geometric distribution, as getting red ball can be termed as success]

Subtracting second from first we get,

   [ Infinite GP series]

So, expected number of draws to see red ball is 5.

(b)

Suppose, random variable Y denotes number of draws required to get red ball.

As we draw without replacement, Y can take any value from 1 to 5.

P(Y=y) depends on each draw of 1,2,...,y. As ball is not replaced, a ball other than red must be drawn out of remaining in each earlier steps.

So, expected number of draws to see red ball is 3.

2.

Suppose there are n blue balls in the jar.

So, in total n+1 balls.

Balls are drawn with replacement.

So, as of 1(a) we can calculate and conclude that expected number of draw is n+1 to get the red ball.

But, it is given that n+1=20.

So, maximum likelihood estimator of number of blue balls in the jar,