Black and white marbles probability question

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In summary, the conversation discusses two-marble and three-marble probabilities of selecting a black marble. Adding a black marble to the two-marble bag changes the probability to 2/3, but this does not reveal the colors of the initial two marbles. The statement that three marbles with 2/3 probability of selecting black must mean two black and one white is incorrect. There are other possible combinations.
  • #1
bahamagreen
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TL;DR Summary
Bag with two marbles, each p=.5 of being black or white. After adding a third black marble, what is p of selecting a black marble from the bag? I get 2/3 which impossibly reveals the two marbles' colors. Not finding the flaw...
Two-marble probabilities
BB 1/4
BW 1/2
WW 1/4

Three-marble probabilities of selecting a black
B BB 1
B BW 2/3
B WW 1/3

Probability of selecting a black marble if we add a black marble to the two marble bag
(1 x 1/4) + (2/3 x 1/2) + (1/3 x 1/4) = 2/3

Three marbles with 2/3 probability of selecting black must mean two black and one white. If I remove (mentally) the black marble I added, I impossibly "know" the colors of the initial randomly selected two-marble bag (although I'm only right half the time).
 
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  • #2
bahamagreen said:
Summary:: Bag with two marbles, each p=.5 of being black or white. After adding a third black marble, what is p of selecting a black marble from the bag? I get 2/3 which impossibly reveals the two marbles' colors. Not finding the flaw...

Three marbles with 2/3 probability of selecting black must mean two black and one white.
This is not correct. 2/3 probability of selecting black could also mean a 1/4 probability of one black and two white plus a 1/2 probability two black and one white plus a 1/4 probability of three black.
 
  • #3
I mean selecting one marble and it being black...
 
  • #4
Pretty trivial to just jot down the very small number of possible situations and it will be obvious what the answer is.
 
  • #5
bahamagreen said:
I mean selecting one marble and it being black...
Yes, understood
 
  • #6
Dale said:
Yes, understood
OK, thanks, been suspecting the "must" in that statement was too exclusive.
 

1. What is the probability of drawing a black marble on the first try?

The probability of drawing a black marble on the first try is 1/2 or 50%. This is because there are 10 marbles in total, with 5 being black and 5 being white, so the chances of drawing a black marble are equal to the chances of drawing a white marble.

2. What is the probability of drawing a white marble on the second try if a black marble was already drawn on the first try?

The probability of drawing a white marble on the second try after a black marble was drawn on the first try is 4/9 or approximately 44.44%. This is because after the first draw, there are 9 marbles left, with 4 being white and 5 being black. So the chances of drawing a white marble on the second try are 4 out of 9.

3. What is the probability of drawing a black marble on the third try if a black marble was already drawn on the first two tries?

The probability of drawing a black marble on the third try after a black marble was drawn on the first two tries is 3/8 or 37.5%. This is because after the first two draws, there are 8 marbles left, with 3 being black and 5 being white. So the chances of drawing a black marble on the third try are 3 out of 8.

4. What is the probability of drawing a white marble on the third try if a black marble was drawn on the first try and a white marble was drawn on the second try?

The probability of drawing a white marble on the third try after a black marble was drawn on the first try and a white marble was drawn on the second try is 2/7 or approximately 28.57%. This is because after the first two draws, there are 7 marbles left, with 2 being white and 5 being black. So the chances of drawing a white marble on the third try are 2 out of 7.

5. What is the probability of drawing a black marble on the fourth try if a black marble was drawn on the first three tries?

The probability of drawing a black marble on the fourth try after a black marble was drawn on the first three tries is 1/6 or approximately 16.67%. This is because after the first three draws, there are 6 marbles left, with 1 being black and 5 being white. So the chances of drawing a black marble on the fourth try are 1 out of 6.

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