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'll call the people 1,2,3 for simplicity and the colors ABCDE so the starting position can be written as
ABCDE
ABCDE
ABCDE
When person 1 pulls, they could get either the same or different colors from 2&3.
Case 1: Same colors p=1/5 (call the color E)
ABCDEEE
ABCD
ABCD
Person 2 must draw an E back from 1 p=3/7 and can draw any from 3 (call it D)
ABCDEE
ABCDED
ABC
Person 3 must draw E back from 1 and D back from 2. (2/6*2/6)=1/9.
Thus the probability of this case is 1/5*3/7*1/9=3/315
Case 2: Person 1 pulls two different colors 4/5 (call them E and D)
ABCDEDE
ABCD
ABCE
The final position can only be restored if person 2 draws either E or D back from 1.
Case 2a. E from 1 any from 3. p=2/7
ABCDDE
ABCDEA
BCE
Person 3 restores: 2/6*2/6=1/9
Final probability 4/5*2/7*1/9=8/315
Case 2b. is symmetric to 2a. so p=8/315
The final probability is thus (3+8+8)/315 = 19/315
Assuming no errors of reasoning.
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Posted by Jer
on 2022-05-07 09:20:51 |
Solution by cases
