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let there be a urn with: $3$ red balls, $4$ white balls, $2$ black balls.
one ball is picked up and returned to the urn, over and over, what is the probability that the first red ball that will be picked, will be before the first white ball.

It is said that we can neglect the black balls, so the answer is $\frac{3}{7}$, why does the black ball can be neglected?

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  • $\begingroup$ because it doesn't affect the order that the red and white balls are pulled out $\endgroup$ – danimal May 18 '15 at 11:36
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You can neglect it, since the task is only interested in the order of the red and white balls, if you pick a black ball, that doesn't effect anything in that case(You can count picking black as the null element).

For example, if you pick a red ball 2 times, and then a black ball 2 times, and then a white ball 2 times $\rightarrow RRBBWW$, that will be the same as you drew red ball 2 times, and white ball 2 times, since the task is only interested in their order$\rightarrow RRBBWW=RRWW$

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Suppose you are throwing coins and asking what is the probability of heads coming before tails.

Now let's change that questions so that you throw the coins and say what was tossed, heads or tails, and the question is what did you say first, heads or tails.

Now let's add to the second variant that every minute you also say what the time is. So you are now saying "heads", "tails" or the time. Saying the time will not change the probability of the heads or the tails. Similarly, the fact that there are black balls and you now say that you picked one out, will not change anything. The question is, what will you say first "red" or "white".

In other words, the question is what is the probability that out of the red and white balls the first one you select is red. That is 3 possibilities out of 7.

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