Probability without replacement

GRE Quant Problem Solving
1

when pulling marbles or anything out of a bag, and the following are given

  • 2 types are in the bag
  • equal amounts of both types are in the bag

isn’t the probability going to change as soon as the first pull occurs, and keep changing each pull? and wouldn’t multiple cases per pull need to be considered?

example:
red and blue marbles in a bag
5 of each are in the bag
what are the odds of blue being pulled on the nth draw

case 1:
blue is pulled first
now the odds of blue getting pulled second are 4/9,

case 2:
red was pulled first. Now the odds of pulling blue second would be 5/9

and so on and so on after each pull.

wouould it be correct to say that the odds are likely not 1/2 the further you get from the first pull?

2

Yes, exactly!

Given there is no replacement, the cases would indeed need to be considered, and yes–there is no case where 50% is an option.

Case 1: Red then Red → 5/10 * 4/9 = 4/18

Case 2: Red then Blue → 5/10 * 5/9 = 5/18

Case 3: Blue then Red → 5/10 * 5/9 = 5/18

Case 4: Blue then Blue → 5/10 * 4/9 = 4/18

It is actually an understatement to say that the odds are unlikely to be 1/2 on the second pull; we’ve shown that it’s impossible!

However, if you keep picking, and assuming that different colors have been selected after 2 pulls, there will be chances for 50% odds again.

Case in Point: Red then Blue leaves 4 of each color remaining, so the odds of selecting blue again are 4/8 or 50%

3

Thanks Connor!

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