Kim has 5 pairs of shoes; each pair is a different color

Question

Kim has 5 pairs of shoes; each pair is a different color. If Kim randomly selects 2 shoes without replacement from the 10 shoes, what is the probability that she will select 2 shoes of the same color?

(A) 2/5 
(B) 1/5 
(C) 1/9 
(D) 1/10 
(E) 1/25

ENGRTOMBA2018 · Accepted Answer

Can be tackled in this way as well: Probability of selecting any 1 out of 10 shoes = 10/10 = 1

Probability of selecting the next shoe (out of 9 available) having the same color = 1/9 (as after selecting the 1st one, there is only 1 another shoe left with the same color).

Thus the total probability = 1*1/9=1/9.

C is the correct answer.

BrentGMATPrepNow · Answer

Here's an approach that uses  counting techniques

P(matching pair) = ( number of ways to get a matching pair )/( TOTAL number of ways to select 2 shoes )

number of ways to get a matching pair  
There are 5 different colors. 
So, there are  5  different ways to get a matching pair

TOTAL number of ways to select 2 shoes  
There are 10 shoes altogether. 
Since the order in which we select the 2 shoes does not matter, we can use combinations. 
We can select 2 shoes from 10 shoes in 10C2 ways 
10C2 =  45

Aside: If anyone is interested, we have a video on calculating combinations (like 10C2) in your head. (see below)

So, P(matching pair) =  5 / 45 
= 1/9
= C

RELATED VIDEO
 https://www.youtube.com/watch?v=05a0A7vjG8I

NoHalfMeasures · Answer

Total pairs = 10C2 = 45; Same Color Pairs= 5C1*1C1=5; Prob = 1/9
Or 2/10 * 1/9 * 5 = 1/9

Ans C

BrentGMATPrepNow · Answer

P(matching pair) = P(select ANY shoe for 1st selection   AND   select matching shoe for 2nd selection)
= P(select ANY shoe for 1st selection)   x   P(select matching shoe for 2nd selection)
= 1   x   1/9
= 1/9
= C

ASIDE: Once we have selected ANY shoe as the 1st selection, there are 9 shoes remaining. Of those 9 remaining shoes, only 1 matches the first shoe (thus the 1/9)

Cheers,
Brent

karan12345 · Answer

Probability of selecting 1 out of 10 shoes = 10/10 = 1
Probability of selecting the next shoe having the same color = 1/9 (as only 1 shoe is left of the same color and 9 shoes are left after selecting one)
So, Probability that she will select 2 shoes of the same color = 1 * 1/9 = 1/9
Hence, Answer choice is (C).

ProfChaos · Answer

Can someone please help me with what is wrong with my approach:

Picking up first shoe: 1/5 (we can pick up any one of the shoe from a pair)
Picking up the 2nd shoe: 1/9 (since only one shoe of the pair is left to be picked up from the remaining 9 shoes)

Reqd Probability = 1/5 * 1/9 = 1/45

BrentGMATPrepNow · Answer

We can pick ANY shoe for the first shoe. So the P(selecting the first shoe) = 1 (not 1/5)

If we say P(selecting the first shoe) = 1/5, then we're also saying that P(NOT selecting the first shoe) = 4/5, which makes no sense.

ProfChaos · Answer

Oh yes!
I understood what you are saying.

So I can pick any shoe in the first pick and thus prob is 1
But the second pick has to be the the other shoe of the pair, hence prob of picking up a second shoe so that both form a pair is 1/9

I had read your explanation before as well but I dont know why, at that time it didnt click with me.
Anyway, thank you so much for your reply  BrentGMATPrepNow

Sneha2021 · Answer

BrentGMATPrepNow

Why numerator is 5c1 instead of 10c1*1?

siddhantvarma · Answer

When you use 10C1*1, you're selecting one shoe from 10 shoes first and then selecting it's matching pair. Let's say you end up selecting a matching pair of blue shoe, say B1 and B2.
10c1*1 will give us a pair B1,B2 and B2,B1 so you're effectively duplicating the number of selections because you're considering their arrangements as different pair. But is that the case? Does it matter if you select B1 first and then B2 or B2 first and then B1?
No, it doesn't matter. You only need to select a pair, so B1,B2 is same as B2,B1
So if you do 10c1*1, you need to divide it by 2 so that you don't consider their arrangements that lead to twice the selections.

Kim has 5 pairs of shoes; each pair is a different color

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Kim has 5 pairs of shoes; each pair is a different color

Prep Toolkit

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