Suppose a box contains 5 white balls and 5 black balls.
If you want to extract a ball and then another:
What is the probability of getting a black ball and then a black one?
I think that this is the answer:
Let $A:$ get a black ball in the first extraction, $B:$ get a black ball in the second extraction. Then: $P(A)=\frac{5}{10}$ and
$$P(B|A)=\frac{P(A\cap B)}{P(A)}=\frac{\frac{4}{9}}{\frac{1}{2}}=\frac{8}{9}$$
Is this correct?
Now, If you want to extract two balls at the same time:
what is the probability that both are black?
Is exactly the same probability of the first question?
Why?
Thanks for your help.
Best Answer
It is good to check an answer for plausibility. Given that the first ball is black, surely the probability that the second ball is black cannot be $\dfrac{8}{9}$!
In this case, it is the conditional probability that is obvious. Note that $\Pr(B|A)=\dfrac{4}{9}$. And what we want is $\Pr(A\cap B)$. We get $$\Pr(A\cap B)=\Pr(B|A)\Pr(A)=\frac{4}{9}\cdot\frac{1}{2}.$$
As to your question about two consecutive picks versus two simultaneous picks, there is no effective difference. If Alicia picks two balls from the urn, what difference can it make whether she touches the two balls simultaneously, or with a delay of a millisecond?
Remark: For this kind of problem, using the formal conditional probability machinery is overkill, and may hinder the intuition.