A drawer contains five socks, three red and two blue. If you pick a sock randomly, replace it, and pick another, what is the probability that both socks are red? If you pick two socks randomly without replacing the first, what is the probability that both are red?

Question

anonymous · Answer

In the first case, it is two independent events... very nice.

anonymous · Answer

So to get one sock it is $$
\frac{3}{5}
$$And two get two socks it is $$
\left(\frac35\right)^2 = \frac9{25}
$$

anonymous · Answer

In the second case, after you pull a red sock, probability changes...

anonymous · Answer

So for the first sock it is $$
\frac35
$$But now you have $3-1=2$ red socks and $5-1=4$ total socks
So you get $$
\frac24
$$Probability for second try if you know the first try was red.

anonymous · Answer

Fortunately this 'if' probability is independent from the first sock drawing, which makes the events independent again!
So the combined probability is $$
\frac35 \times \frac24=\frac6{20}=\frac3{10}
$$

anonymous · Answer

Thank you!