A drawer contained red socks and black socks. When two socks are drawn

Question

A drawer contained red socks and black socks. When two socks are drawn at random, the probability that both are red is 1/2. What is the minimum number of socks that could be in the drawer ?

A. 2
B. 3
C. 4
D. 5
E. 6

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ayushv005 · Answer

R = red no. of balls, B = Black no. of Balls, N = Total no. of balls.

Given, R/(R+B)*(R-1)/(R+B-1) = 1/2
 => 2R(R-1) = (R+B)(R+B-1)
 => 2R(R-1) = (N)(N-1) 
 => 2R^2 - 2R = N^2 - N
 => N = 2R
 Since there are at least 2 red balls, min(N) = 2*2 = 4
 (N = 4, R = 3, B = 1 satisfies the given conditions as well)

sanya2711 · Answer

how did you get from the highlighted equation 1 to highlighted equation 2?

sthahvi · Answer

is there a way to solve this by logic ?

Sabby · Answer

Hi  sthahvi ,

You can think of it this way :

R/x * (R-1)/(x-1)=1/2
so basically the denominator must be in the form of x(x-1) and the numerator as well R(R-1)
we could for example say 2*1 but that would mean that we have 2 socks and therefore only one pair of Red and no other socks, however the problem says we also have Black socks

Therefore x must be greater than 2

The next option is x=4 
R/4*(R-1)/3=R(R-1)/12
it gives us R(R-1)=6 so R=3 and R-1=2
6/12=1/2
we have 4 socks minimum, 3 red and 1 black in this case

by the way I intentionally omitted x=3 because even though we can have the denominator in form of 3*2 the numerator can't equal 3 as two consecutive integers can't be odd and we would need a product resulting in 3 to get 3/6=1/2

Let me know if this helps!

Regards,

sthahvi · Answer

thank you this did help

Paras96 · Answer

A drawer contained red socks and black socks. When two socks are drawn at random, the probability that both are red is 1/2. What is the minimum number of socks that could be in the drawer ?

Let no. of red socks = n
Let no. of blacks socks = m

Probability = nC2/(n+m)C2

Among the options, we can check them one by one
A. 2 if total socks = red socks = 2, probability = 1
B. 3 if total socks = 3, Probability = nC2/3C2 =, If n =2, Probability = 2/3
 C. 4 if total socks = 4, Probability = nC2/4C2 =, If n =2, Probability = 2/4 = 1/2, If n=3 Probability 3/4  
D. 5
E. 6

Rest options i.e. D & E are not minimum and therefore answer is C

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A drawer contained red socks and black socks. When two socks are drawn

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