Saul, Simon and Stuart each have a bag containing 5 different colored marbles. Each bag contains the same five colors.
o Saul reaches into Simon's bag and Stuart's bag and randomly withdraws a marble from each and places them in his bag.
o Simon then reaches into Saul's and Stuart's bag and randomly withdraws a marble from each and places them in his bag.
o Finally, Stuart reaches into Saul's and Simon's bag and randomly chooses a marble from each and places them in his bag.
Determine the probability that each of the three boys has five different colors of marbles in their bag.
I'm going to generalize a bit to N different colored marbles.
Saul can either grab two marbles of the same color or two marbles of different colors.
Case 1: Saul grab two marbles of the same color, call that color red. Probability 1/N.
Then to end up with all N different colors Simon must grab one of the three red marbles from Saul; the marble taken from Stuart could be any color other than red, call it blue. Probability 3/(N+2)
Then for Stuart to end up with all N different colors he must grab one of the two red marbles from Saul and one of the two blue marbles from Simon. Probability (2/(N+1))^2
Total probability in this case is 1/N * 3/(N+2) * (2/(N+1))^2 = 12/(N*(N+1)^2*(N+2))
Case 2: Saul grabs two marbles of different colors, call the one from Simon red and the one from Stuart yellow. Probability (N-1)/N.
Simon must grab a red marble or a yellow marble from Saul.
Subcase 2A: Simon grabs a red marble from Saul. Probability 2/(N+2)
Simon's second marble can be any of the remaining marbles from Stuart's bag, call it blue.
Stuart then must grab one of the two yellow marbles from Saul and one of the two blue marbles from Simon. Probability (2/(N+1))^2
Total probability in this case is (N-1)/N * 2/(N+2) * (2/(N+1))^2 = (8*(N-1))/(N*(N+1)^2*(N+2))
Subcase 2B: Simon grabs a yellow marble from Saul. Probability 2/(N+2)
Simon's second marble grabbed must then be the red marble from Stuart. Probability 1/(N-1).
Then Stuart must grab a red marble from Saul and a yellow marble from Saul. Probability (2/(N+1))^2.
Total probability in this case is (N-1)/N * 2/(N+2) * 1/(N-1) * (2/(N+1))^2 = 8/(N*(N+1)^2*(N+2))
Total probability of all cases is 12/(N*(N+1)^2*(N+2)) + (8*(N-1))/(N*(N+1)^2*(N+2)) + 8/(N*(N+1)^2*(N+2)) = (8N+12)/(N*(N+1)^2*(N+2))
evaluating this at N=5 gives a probability of 13/315 for the specific problem.
Solution
