**bag
of marbles**

You have 100 marbles numbered 0 to 99 in a bag. You repeatedly draw a marble from the bag (all marbles being equiprobable), note its number, and replace it in the bag. On average, how many of the marbles numbered 1 through 99 will have been drawn from the bag (one or more times) before drawing marble #0?

__Solution by __**Michael
A. Gottlieb**

The probability of drawing a given marble *m*
(other than 0) before drawing marble 0 must be the same as the
probability of drawing marble *m* for the first time before drawing marble
0 for the first time if you were to continue to draw (and replace) marbles until
every marble (0-99) had been drawn at least once. However, there is nothing
special about marbles *m* and 0 in this regard - the probability must
be the same for any two given marbles. In particular, it must equal the
probability that marble 0 occurs for the first time before marble *m*
occurs for the first time, and therefore it must equal 1/2. The expected number
of marbles that will be drawn before 0 is drawn is just the number of non-0
marbles (99) times the probability that each will be drawn before 0 is drawn
(1/2), which is 99/2.