A drawer contains 5 brown socks and 4 blue socks well mixed. A man reaches the drawer and pulls out 2 socks at random. What is the probability that they will match?

2 Answers
LM
May 30, 2018

#4/9#

Explanation:

before the man takes out any socks, there are #9# socks in the drawer.

#5# socks, and so #5/9# of the socks in the drawer, are brown.
#4# socks, and so #4/9# of the socks in the drawer, are blue.

the probability of pulling out a brown sock at this point is #5/9#, and the probability of pulling out a blue one is #4/9#.

if a brown sock is pulled out and not replaced:

there are #8# socks left in the drawer. #4# are brown and #4# are blue.
#4/8# of the socks left are brown, and #4/8# of the socks left are blue.
the question asks for the probability of a matching sock (here, another brown one). the probability of pulling out a brown sock at this point is #4/8# or #1/2#.

pulling out #2# matching brown socks means pulling out one brown sock #(5/9)# and pulling out another brown sock #(4/8)#.

the probability of both happening together is found by multiplying the two probabilities.
#5/9 * 4/8 = 20/72#
hence, the probability of pulling out #2# brown socks is #20/72#.

if a blue sock is pulled out and not replaced:

#5/8# of the socks left are brown, and #3/8# of the socks left are blue.
the probability of pulling out a blue sock at this point is #3/8#.

pulling out #2# matching blue socks means pulling out one blue sock #(4/9)# and another blue one #(3/8)#.

again, the probability of both happening together is found by multiplying the two probabilities.
#4/9 * 3/8 = 12/72#
hence, the probability of pulling out #2# blue socks is #12/72#.

meanwhile, getting matching socks can happen in #2# ways: matching brown socks or matching blue socks.

to find the probability of either one happening (note that they can't both happen together), you can add the probability of each one happening.

#20/72 + 12/72 = 32/72#
this can be simplified to #4/9#.
hence, the probability of pulling out #2# matching socks (of either colour) is #4/9#.

Mark D.
May 30, 2018

#4/9#

Explanation:

He can pick brown brown
Or brown blue
Or blue blue
Or blue brown

Only the first and third choices are what we want

Brown is #5/9# then #4/8# as it is not replaced

Blue is #4/9# then #3/8# same reason

So P( 2brown)#=5/9xx4/8=20/72#

P(2 blue)#=4/9xx3/8=12/72#

P(2 same)#=20/72+12/72=32/72=4/9#

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