I was going through the ‘Intro to probability’ quiz on Prepswift.
I know that in ‘with replacement’ the probability of drawing a marble of the same color in a bag of marbles is the same in each draw (first, fifth, or n-th). However, in ‘without replacement’ the number of marbles decreases and the probability changes in each draw, making the earlier draws with more probability than the later draws. I am not sure about the answer provided in the quiz.
Screenshots attached
See this smaller example: https://www.reddit.com/r/GRE/comments/17v5396/gregmat_quant_problem_if_i_take_out_a_red_third/
It’s the same concept here.
i am still a little confused here and dont understand why reducing the number of marbles serves as a sufficient explanation?
for case 1 - selecting the red marble third:
if no red is selected in prior two draws, max probability is 17/48 for third draw. if two reds are selected in first two draws, min probability is 15/48.
for case 2 - selecting the red marble tenth:
if no red is selected in prior 9 draws, max probability is 17/41 for tenth draw. if all 9 prior draws selected all reds, min probability for tenth draw is 8/41.
now i see quantity A being between 17/48 and 15/48 or roughly 0.35 > A > 0.31,
and quantity B is between 17/41 and 8/41 or roughly 0.41 > B > 0.195
with those ranges, A could be bigger than B, or B could be bigger than A , depending upon what gets chosen in the prior positions.
now, one caveat is that i am assuming these situations are independent, which may be where i go wrong?
but if the maximum for A can occur within the same selection of the first 10 as the maximum of B, and the minimum of A can occur within the same selection of the first 10 as the minimum of B, i don’t understand how you can know for sure that these selections have the same probability without already knowing for sure what was selected during the 1,2,3,4,5,6,7,8,9 selections.
overall i am just confused how they are equal and would appreciate a more in depth explanation or at least proof as to why my reasoning above is wrong.
i should have reviewed conditional vs unconditional probability cases before posting my reply. so when a probability problem on the GRE doesn’t specify conditioning information, we are safe to interpret it as asking for the unconditional probability?
Well, nothing is restricting you from picking a red marble in the first or second pull as well.
Likewise here
It also looks like you’re stopping after the “third” or “tenth” pull, which should still give you the same answer. However, for visualization purposes, it’s better to conform to what the question suggests : “randomly pull marbles one by one,” implying that all marbles are pulled.
Try using the above clarification to construct some kind of “proof” using combinatorics
