Question 572445: You have 2 jars, 10 red marbles, 40 blue marbles. you need to place all the marbles into the jars such that when you blindly pick one marble out of one jar, you maximize the chances that it will be red. (when picking, you'll first randomly pick a jar, and then randomly pick a marble out of that jar) you can arrange the marbles however you like, but each marble must be in a jar.
Q1) What is the best to maximize the chances to pick a red marble?
Q2) What is the probability of picking a red marble in your answer?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! if you put 10 red marbles in 1 jar and 40 blue marbles in the other jar, then the probability of getting a red marble is 50% because 50% of the time you will be picking 1 jar and 50% of the time you will be picking the other jar.
suppose you put 1 red marble in 1 jar and 9 red marbles in the other jar,
then the probability would be 50% * 1 + 50% * 9/49 = .5918367347 because 50% of the time you are picking from 1 jar and 50% of the time you are picking from the other jar.
if you put 2 red marbles in 1 jar and 8 red marbles in the other jar, then the probability would be 50% * 1 + 50% * 8/48 = 583333333
looks like it goes downhill from there.
the more red marbles you put in the second jar, the lower the probability of getting a red marble out of either jar.
the probability of getting a red marble out of the first jar is always 1 so that remains a constant.
the less red marbles you put in the other jar, the lower the probability for that jar because:
9/49 is greater than 8/48 is greater than 7/48 is greater than 6/47, etc.
in terms of weighting, 1 red marble worth 50% carries more weight than 2 red marbles at 50%.
bottom line is:
1 red marble in 1 jar and 9 red marbles in the other jar added to the 40 that are already there.
the probabilities listed in order are:
(0 + 10/50) / 2 = .2
(1/1 + 9/49) / 2 = .59183673 *****
(2/2 + 8/48) / 2 = .58333333
(3/3 + 7/47) / 2 = .57446809
(4/4 + 6/46) / 2 = .56521739
(5/5 + 5/45) / 2 = .55555556
(6/6 + 4/44) / 2 = .54545455
(7/7 + 3/43) / 2 = .53488372
(8/8 + 2/42) / 2 = .52380952
(9/9 + 1/41) / 2 = .51219512
10/10 + 0/30) / 1 = .5
the highest probability is when you put 1 red marble in 1 jar and the rest of the marbles in the other jar.
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