Question

There are 10 table-tennis balls in a box. One of them has “WIN” written on it; and the others are numbered 1 through 9. You select (randomly) a ball. If the ball is numbered N, you put it back into the box and wait for N minutes. Then, you select (randomly) a ball, again, and repeat, waiting for that many minutes as written on each ball, until your selection is the ball labeled “WIN”. End of the game… What is the expected duration of the game (in minutes)? (If you win in your first selection, the elapsed time is zero.)

Please solve the question step by step with an explanations and details.

The correct answer is 45 not 4.5. Be careful since drawn ball put back into the box, it is possible that 'win' written ball can be drawn after more than 10 even 20 drawing.

Answer 1

Let the probability of success i.e. getting the WIN ball is = p = 1/10

Therefore, the probability of getting any other ball = failure = q = 9/10

Probability of getting first success after x failure is given by

for x = 0 to infinity

So for the above distribution E(X) = 1/q = 9

However, according to the problem, a person needs to wait N minutes according to number written on the ball drawn.

We define the following

Y_i = Time after number i numbered ball is drawn, where i = 1 to 9

P(Yi) = 1/9

The required expectation of wait time