There are 100 boxes that each contain a $1 bill. You choose a random box, take t
ID: 3044284 • Letter: T
Question
There are 100 boxes that each contain a $1 bill. You choose a random box, take the bill (unless the box is already empty), then replace the bill. You do this 50 times.
a) How likely is it that a given box will be chosen at least once?
b) How much money do you expect to win?
For a), my intuition is that the probability would be 1/2 if the boxes weren't replaced, so it should be less than 1/2 with replacement. For part b) I know the expected winnings is between 0 and 50 noninclusive, but don't know how to find the probability of winning any certain amount (i.e., P($20)).
Explanation / Answer
probability of opening a particular box in a turn =1/100 (as there are 100 boxes)
hence probability of not opening a particular box in a turn =1-1/100 =99/100
a)
probability that a given box will be opened at least once during a game =1-P(it will not get opened in 50 turns)
=1-(99/100)50 =1-0.6050 =0.3950
b) expected win from 1 box =$1 * probability that it gets opened at least>
hence expected win from 100 box =100*0.3950 =$39.50
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