Question: You have two black socks, three white socks and five brown socks in your suitcase. it is dark and you pick out three socks from the suitcase without looking at the colour. find the probability that you do not pick a pair of the same colour?
Solution:
Given, black socks = 2
white socks = 3
brown socks = 5
Let E be the event that All chosen socks are of the same colors
$P(E↖{-}) = 1 - P(E)$
= 1 - (all are white or all are brown)
$= 1 - ( 3/10 x 2/9 x 1/8 + 5/10 x 4/9 x 3/8)$
$= 1 - (1/{120} + 1/{12})$
$= 1 - {1 + 10}/120 $
$= 1 - {11}/{120}$
$={120-11}/120$
$= {109}/{120}$