A box contains $10$ pairs of shoes ($20$ shoes in total). If two shoes are
selected at random, what it is the probability that they are matching
shoes?
I saw people answering this question with the answer of $\frac{1}{19}$.
But I don't understand why we cannot also reason that the two shoes (that are a pair) are picked at the same time out of the $20$ shoes, hence it's $\frac{2}{20}$ or $\frac{1}{10}$.
Best Answer
If you first pick a shoe, then the probability that the next shoe you pick comes from the same pair as the first shoe is $1/19$.
The number of ways two shoes can be selected from $20$ shoes is $$\binom{20}{2}$$ Ten of these choices form a matching pair, so the number of favorable cases is $10$. Hence, the probability of selecting a matching pair of shoes is $$\frac{10}{\binom{20}{2}} = \frac{10}{190} = \frac{1}{19}$$ To get an answer of $1/10$, you would have to set the rules so that you always pick a left shoe and a right shoe, as Mark Bennet indicated in the comments.