Probability concepts - how can balls of same colour be distinguishable?

Question

An urn contains $6$ white and $4$ black balls. A fair die is rolled and that number of balls are chosen from the urn. Find the probability that the balls selected are white.

I know the basic way to go about solving the problem.

Let $W$ be the event of finally drawing all white balls. Let $P(n)$ denote the probability of appearance of $n$ on the die.

We want: $$P(W) = P(1)P(W\mid 1)+ P(2)P(W\mid 2)+\dots \implies P(W) = \dfrac{1}{6}\left(\sum_{i=1}^6P(W\mid i)\right)$$

Now, I am actually facing trouble in computing $P(W/i)$. I saw author's method and in it he has used $P(W\mid i) = \dfrac{^6C_i}{^{10}C_i}$

but I fail to understand how that can be true when all white balls are identical and all black balls are identical.

Here, $^6C_i$ denotes the combination of $i$ different things from 6 different objects, doesn't it? How can that be used here?

Answer 1

• The probability of drawing $i$ white balls is the probability of drawing a white ball and then (without replacement) drawing a second white ball, and so on up to $i$

• The first ball drawn has a probability of $\frac{6}{10}$ of being white

• Given that the first ball drawn is white, the second ball drawn has a probability of $\frac{5}{9}$ of being white

• An so on up to the $i$th ball having a probability of $\frac{6-i+1}{10-i+1}$ of being white

• So the probability all $i$ balls drawn are white is $$\frac{6 \times 5 \times \cdots \times(6-i+1)}{10 \times 9 \times \cdots \times(10-i+1)} = \dfrac{\frac{6!}{(6-i)!}}{\frac{10!}{(10-i)!} }= \dfrac{\frac{6!}{(6-i)!i!}}{\frac{10!}{(10-i)!i!}} = \dfrac{^6C_i}{^{10}C_i}$$

Answer 2

If you draw the $i$ balls sequentially $$P(W\mid i)=\frac{6}{10}\cdot\frac{5}{9}\cdots\frac{6-i+1}{10-i+1}$$ If you draw the $i$ balls simultaneously $$P(W\mid i)=\frac{\dbinom{6}{i}\cdot\dbinom{4}{0}}{\dbinom{10}{i}}$$ Now, can you convince yourself that these two are equivalent?

Answer 3

Since I don't see it anywhere, let me address this part of your question, rather than specific case:

how that can be true when all blue balls are identical and all black balls are identical.

Here, $^6C_i$ denotes the combination of $i$ different things from 6 different objects, doesn't it? How can that be used here?

Let's make a simple thought experiment. Let's say that rather than having indistinguishable white and black balls you have 10 balls numbered 1 to 10. Balls numbered 1 to 6 are white and those numbered 7 to 10 are black.

Now you have 6 white balls and 4 black balls that are distinguishable. How does that change your probability?

The numbers on balls change nothing in probability as long as the only thing you're concerned is ball colour. The resulting probability has to be the same as if there were no numbers. We may simply ignore the numbers on the balls and follow the original problem. We still have 6 white balls and 4 black balls so the old approach holds. Whatever other way you count the probability it has to provide the same result.

But now your balls are distinguishable so you can apply methods specific to distinguishable balls, specifically use combination. The results, as shown earlier will be the same.

This is why to indistinguishable balls you can always apply approach "Let's assume the balls are distinguishable..."