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Chips from a box

ProbabilityEasy

A box holds $5$ red chips and $3$ blue chips. You draw $2$ chips one after another without replacement, each remaining chip being equally likely. Given that the second chip is blue, what is the probability that the first chip is also blue? Give a fraction in lowest terms.

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Hint

Once you know the second chip is blue, count how the first chip can look among the other $7$ chips.

Solution

Condition on the second chip being blue, so the first chip is one of the other $7$ chips by symmetry.
Among those $7$ chips exactly $2$ are blue, because one of the $3$ blue chips is the second draw.
By symmetry every one of the $7$ chips is equally likely to be the first draw.
So the probability is $\frac{2}{7}$ .

Answer: $\frac{2}{7}$

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