# Thread: Would love some assistance with understanding this puzzling probability "game"

1. ## Would love some assistance with understanding this puzzling probability "game"

Hello everyone,

This is my first post here, so please be gentle .

I'd like to thank everyone in advance for any help that can be given.

I'm having great difficulty understanding the "logic" behind a certain probability problem. It takes the form of an imaginary game. While I'm sure my math is correct (I even wrote a simple computer program to simulate / compile 1 million trial runs of this game, and it matched my calculated probabilities perfectly), for the life of me I just can't understand why the result doesn't make logical sense.

This is not a homework problem; I'm a Ph.D. Organic Chemist (38 years old), and this is just something that's been nagging at the back of my mind since undergrad.

So, the problem takes the form of an imaginary game. You have a cloth bag with ten marbles inside. The marbles are either white or black, but there must be at least one white and one black marble in the bag. Acceptable combinations are 9 white and 1 black, going down to 1 white and 9 black.

The rules of the game are that you reach in the bag and draw out a marble. If it's a white marble, you 'win', and the game stops. If it's a black marble, you put it aside and continue to draw marbles out of the bag one at a time until you pull a white marble, at which point you've won.

So I begin to calculate the probabilities. First, with 9 white marbles and 1 black marble:

Probability of winning on the first draw:

(9/10) = 90%.

Probability of winning on the second draw:

(1/10) * (9/9) = 10%. This makes sense to me, that it's more difficult to win on the second draw. You have to avoid hitting all those white marbles and pluck out the single black marble, followed by pulling out one of the remaining marbles (all white, 9/9) to end the game.

So far everything is great. If you continue the math on down the line (8 white 2 black, 7 white 3 black, 4 white 6 black, etc), the trend continues. It becomes more and more difficult to fish out all the black marbles without hitting a white one. This is similar to the above example, where it's much harder (10% vs 90%) to avoid hitting a white marble and ending the game prematurely.

With all possible white/black combinations, the math makes sense and this described behavior continues, and it makes perfect sense to me. However, the problem comes when there is only 1 white marble, and 9 black marbles.

Chance of winning on the first draw: 1/10, or 10%.

Chance of winning on the second draw: (9/10) * (1/9), or 10%.

Chance of winning on the third draw: (9/10) * (8/9) * (1/8), or 10%.

...

Chance of winning on the 10th draw: I won't draw it all out, but it's also 10%.

So unlike all the other examples, where it becomes harder and harder to avoid the white marbles as you continue to just hit the blacks, with this combination the trend flatlines. Even though you have to continue missing that one white marble for many consecutive draws - something that in other white/black ratios was shown to be difficult to do - here, it's an equal probability of all possible results.

I just can't wrap my head around the logic of this. If anyone could help explain this behavior, I'd be very much gratified, because this has been driving me crazy since the last century!

2. ## Re: Would love some assistance with understanding this puzzling probability "game"

hi,
I think in general you can have a formula thta would look like this: assuming we have w white balls and N balls in total ( that is N-w blacks, the probability of winning at the first draw will be P(1)= w/N .

P(2) will be ((N-w)/N )*w/(N-1)

P(3) will be ((N-w)/N)*((N-w-1)/(N-1))*w/(N-2) and so on.

Interestingly if w=1 you can simplify the products and you get P(k) = 1/N for all k.

regards
rogojel

3. ## The Following User Says Thank You to rogojel For This Useful Post:

Habanero (07-05-2014)

4. ## Re: Would love some assistance with understanding this puzzling probability "game"

Consider not stopping after you get the white marble. If you do this then instead of "black black white" you'll consider your experiment outcome to be "BBWBBBBBBB" where B represents black and W represents white.

Now - how many possible outcomes are there? Only 10 - one for each of the spots that the white marble could be in. Since we're looking at complete drawing all of the marbles each of these sequences has an equal probability. So each position has exactly one outcome where the white marble was drawn in that position (out of the ten possible). So there must be a probability of 1/10 for each of the positions.

5. ## The Following User Says Thank You to Dason For This Useful Post:

Habanero (07-05-2014)

6. ## Re: Would love some assistance with understanding this puzzling probability "game"

Thanks to everyone for their very nice replies. I see the logic behind the results, now.

I guess my problem was that most of the times, trends continue. Just in general, in life, etc. I'm certainly aware of times when it doesn't, but in the more 'pristine' world of math (as compared to chemistry, which is my home), I was really unprepared for the result where everything is 10%. Everything up to that point had made "sense" to me, but then the perfectly correct result of 10% jarred up against my expectations.

Shame on me . Molecules have been proving my intuitions wrong for 20 years, and I shouldn't expect a different field of study to be any different. Sometimes looking for a trend / expecting a trend to continue indefinitely, instead of concentrating solely on the results of the experiment / calculation, can lead you astray.

Thanks again to everyone who helped me with this. Have a great week!

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