Amy has only two types of socks red and blue She has no idea

Amy has only two types of socks: red and blue. She has no idea how many socks she has, but she does know that if she randomly grabs two socks out of her dresser drawer she has a 1/2 chance of pulling out two red socks.

a. What is the smallest number of socks she could possibly have in her dresser drawer?

b. If there is an even number of blue socks, what is now the smallest possible number of socks in her drawer?

Solution

Amy has only two types of socks

Prob for drawing 2 red socks = 1/2

Let r red socks and b blue socks be there

Prob for drawing 2 red socks = rC2/(r+b)C2 = 1/2

Or r(r-1)/(r+b)(r+b-1) = 1/2

Hence 3C2/4C2 = 1/2

Thus 3 red socks and 1 blue socks she has

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b) if b is even, then

prob =1/2 implies

rc2/(r+b)C2 = 1/2 where b is even

rC2/(r+2k)C2 = 1/2 = 2/4

There are no integers satisfying this. Hence no solution