M02-06

Question

A drawer contains 5 pairs of white socks, 3 pairs of black socks, and 2 pairs of grey socks. If four individual socks are randomly chosen without replacement, what is the probability of getting at least two socks of the same color?

A. \(\frac{1}{5}\)

B. \(\frac{2}{5}\)

C. \(1\)

D. \(\frac{3}{5}\)

E. \(\frac{4}{5}\)

A.  \frac{1}{5}

B.  \frac{2}{5}

C.  1

D.  \frac{3}{5}

E.  \frac{4}{5}

Bunuel · Accepted Answer

Official Solution:

A drawer contains 5 pairs of white socks, 3 pairs of black socks, and 2 pairs of grey socks. If four individual socks are randomly chosen without replacement, what is the probability of getting at least two socks of the same color?

A. \(\frac{1}{5}\)

B. \(\frac{2}{5}\)

C. \(1\)

D. \(\frac{3}{5}\)

E. \(\frac{4}{5}\)

A.  \frac{1}{5}

B.  \frac{2}{5}

C.  1

D.  \frac{3}{5}

E.  \frac{4}{5}

There is no need for a formula to answer this question. The key point is that we have only three distinct colors of socks, yet we are selecting four socks. This guarantees that we will have at least one pair of socks of the same color, regardless of which socks we select. If the first three socks are of different colors, then the fourth sock must match one of the previously selected socks. Therefore, the probability of getting a matching pair is 1.

Answer: C

XyLan · Answer

Given Data:  
 10  W hite | 6  B lack | 4  G rey

JIAA  By the complement rule:
  Total = Case-1 ( when probability is at least 2 ) + Case-2 ( When probability is less than 2 )

To  purposefully  find the probability for which we get  less than 2  socks of the  same color , Let's  fill  the 3 of the 4 empty spaces:
 W - B - G -  __  4th-space  __

As we can see, the  4th-space  can  ONLY  be filled by  any  of the  already  existing colors of the socks , i.e., by  W hite,  B lack OR  G rey, thus  NO  scenario is possible that can fit the conditon.
 Hence, the probability for which we get  less than 2  socks of the   same color   =   0     - The Complementing Case

Therefore,  ALL  the possible cases have at least 2 socks of the same colors, i.e.,  Probability = 1

udayad · Answer

Dear Sir,
If there are 10 white, 6 black and 4 green socks and if 4 socks are picked up individually, then for instance all the 4 may be white, which makes the probability of at least 2 same colored socks as 0.

Bunuel · Answer

If there are 4 white socks aren't at least 2 of them same color, white?

gbernad · Answer

I think this is a  high-quality  question and I agree with explanation.

PerseveranceWins · Answer

I think this is a  high-quality  question and I agree with explanation.

Chalat · Answer

I think this is a  high-quality  question and I agree with explanation.

JIAA · Answer

Hi  Xylan

I understand logic works perfectly here!

What about complement rule??

THANKS IN ADVANCE

Godsent · Answer

This is a good and tricky question.

Posted from my mobile device

RenanBragion · Answer

I think this is a  high-quality  question and I agree with explanation.

gmatbd · Answer

I think this is a  high-quality  question and I agree with explanation.

Ea306 · Answer

I think this is a  high-quality  question and I agree with explanation.

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

sanchitb23 · Answer

I think this is a  high-quality question  and I agree with explanation.

hydr01 · Answer

If we were to use a formula, what formula would we use? Would the solution be a mix of combinations and probability formulas? Can you please walk through what using a formula would look like? If it's easier, what formula would we use if there there a fourth color pair of socks in the mix?

Bunuel · Answer

WWWWWWWWWW - BBBBBB - GGGG
 
We need to find the probability of the following scenarios:

1. Two socks of one color and one sock each of the other two colors (XXYZ)
WWBG
BBWG
GGWB

2. Three socks of one color and one sock of another color (XXXY)
WWWB
WWWG
BBBW
BBBG
GGGW
GGGB

3. Two socks of each of two colors (XXYY)
WWBB
WWGG
BBGG

4. Four socks of the same color (XXXX)
WWWW
BBBB
GGGG

Calculating the probabilities of each scenario separately would be tedious. Instead, we can find the probability of the complementary event (i.e., selecting four socks with no two socks of the same color) and subtract it from 1. However, it is not possible to draw four socks with no two socks of the same color, since we only have three colors. Therefore, the probability of the complementary event is 0.

Thus, the probability of getting at least two socks of the same color is:

P(at least two of the same color) = 1 - P(complementary event) = 1 - 0 = 1.

hydr01 · Answer

Thank you! Also, is the probability of selecting four socks of the same color = ((5/10)*(4/9)*(3/8)*(2/7))+((6/10)*(5/9)*(4/8)*(3/7))+((4/10)*(3/9)*(2/8)*(1/7))?

Bunuel · Answer

10/20*9/19*8/18*7/17 + 6/20*5/19*4/18*3/17 + 4/20*3/19*2/18*1/17.

BottomJee · Answer

I think this is a  high-quality  question and I agree with explanation.

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