Steve has 18 black socks and 12 blue socks Every morning he

Steve has 18 black socks and 12 blue socks. Every morning he pulls a sock at random from his drawer, then pulls a second sock at random, and wears both socks, regardless of their color. What is the probability he wears mismatched socks?

Solution

there are all total 30 socks out of which 18 are black and 12 are blue.

let X be the event that he wears mismatched socks. it means he pulls one black sock and one blue sock.

so P[X]=P[(at first he pulls a black sock,then a blue sock)or(at first he pulls a blue sock, then a black sock)]

          =P[at first he pulls a black sock,then a blue sock]+P[at first he pulls a blue sock, then a black sock] [since the events are mutually disjoint]

          =P[he pulls a black sock from 18 black socks and 12 blue socks]*P[after that he pulls a blue sock from the remaining 17 black socks and 12 blue socks]+P[he pulls a blue sock from 18 black socks and 12 blue socks]*P[after that he pulls a black sock from the remaining 18 black socks and 11 blue socks]

          =18/30*12/29+12/30*18/29=0.4965 [answer]