Probability : A bag contains 12 pair socks . Four socks are picked up at random. Find the probability that there is at least one pair.
My approach :
Number of ways of selecting 4 socks from 24 socks is n(s) = $^{24}C_4$
The number of ways of selecting 4 socks from different pairs is n(E) = $^{12}C_4$
$$\Rightarrow P(E) = \frac{^{12}C_4 }{^{24}C_4}$$
Therefore, the probability of getting at least one pair is
$$1- \frac{^{12}C_4 }{^{24}C_4}$$
But the answer is $$1- \frac{^{12}C_4 \times 2^4}{^{24}C_4}$$… Please guide the error .. thanks..
Best Answer
Suppose the 12 pairs are different.
The number of ways of selecting 4 socks is $24*23*22*21$.
The number of ways of selecting 4 socks of no pair is $24*22*20*18$.
The chances of not getting a pair, then is $18*20 / 23*21$, or $120/161$. (by removing the common factors $22*24$.
The chances of getting at least one pair is then $41/161$
The error in your calculation, is that you are only counting one sock in each pair, similar to the second line of mine being $12*11*10*9$. There are two socks in each pair, so you need to realise that either of the two can be fetched when a pair is selected.