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.