For Cathy's birthday, her uncle decided to make her a deal. He took ten singles and ten one hundred dollar bills, and asked Cathy to divide them into two piles as she saw fit. He would then blindfold her, and thoroughly shuffle each pile of bills, so that the order was completely random. Finally, he would put each pile in a separate box.
Cathy is to pick one of the two boxes at random, and then pick out a random bill from that box (still blindfolded). She would get to keep whatever bill she pulls out.
Naturally, Cathy prefers to get a $100 bill. What strategy should she use in breaking up the bills into two piles to maximize her chance of getting a hundred?
Let us denote the piles as Pile-1 and Pile-2.
Cathy should place precisely one $100 note in Pile-1 and she should put the ten $1 notes and the remaining nine $100 notes in Pile-2.
Now, probability that Cathy picks Pile-n =1/2, where n=1,2
If Cathy picks Pile-1, probability that she will pick $100 =1
Probability that a note picked from Pile-2 is a $100 note
= (# $100 notes)/(total # notes) = 9/19
Consequently, the required probability that Cathy will receive a $100 note
= (1/2)*1 + (1/2)*(9/19)
= (19+9)/38
=14/19 = 0.7368421(correct to seven decimal places)
[EDIT] There was a typing anomaly, which has now been corrected
Edited on January 2, 2022, 11:51 pm
Puzzle Solution
