A box contains 10 pairs of shoes (20 shoes in total). If two

Question

A box contains 10 pairs of shoes (20 shoes in total). If two shoes are selected at random, what it is the probability that they are matching shoes?
1/190

1/20

1/19

1/10

1/9

OA:

Show Spoiler

C

Please see my method:
...................................................

Numerator:
Out of 20 shoes we have to select one shoe first..w/o any restrictions.So that's 20C1 (choosing one out of 20).
Now,the second show has to be the matching shoe.There is only ONE such possible shoe,so the probability of choosing that shoe is 1.

Denominator:
Chosing 2 shoes out of 20.So, 20C2

20C1/20C2 = 2/19

please explain:(

PiyushK · Accepted Answer

probability to choose any shoe out of 20 is 20/20
probability to choose its pair is 1 out of 19 i.e 1/19.

thus (20/20)*(1/19) = 1/19

Bunuel · Answer

You are right it's not the same thing:

First of all:
In numerator you have the number of selection of ONE shoe out of 20, but in denominator TWO shoes out of 20. In my solution both numerator and denominator counting the # of selections of TWO shoes, though from different quantities.

Second, consider this: what is your winning (favorable) probability? You want to choose ONE pair. Out of how many pairs? Out of 10. Now, what are you actually doing? You are taking TWO shoes out of 20, "hoping" that this two will be the pair. So anyway wining scenarios are 10C2 and total number of scenarios are 20C2.

You can do this also in the following way (maybe this one would be easier to understand):
Numerator: 20C1 - # of selection of 1 shoe out of 20, multiplied by 1C1, as the second one can be only ONE, as there is only one pair of chosen one. Which means that # of selection would be 1C1. Put this in numerator.

Denominator would be: 20C1 # of selection of 1 from 20, multiplied be 19C1 # of selection from the 19 left.

So, P=20C1*1C1/20C1*19C1=1/19, here we have in the numerator the same thing you wrote, BUT if we are doing this way then in denominator you should also count the # of selection of the first one out of 20 and the second one out of 19.

The first way of solving actually is the same. Take one shoe from 20, any shoe from 20, I mean just randomly take one. Then you are looking at your 19 left shoes and want to choose the pair of the one you've already taken, as in 19 you have only one which is the pair of the first one you have the probability 1/19 (1 chance out of 19) to choose the right one.

Bunuel · Answer

The problem with your solution is that we don't choose 1 shoe from 20, but rather choose the needed one AFTER we just took one and need the second to be the pair of it. So, the probability would simply be: 1/1*1/19(as after taking one at random there are 19 shoes left and only one is the pair of the first one)=1/19

Answer: C.

We can solve it like you were doing:

P=Favorable outcomes/Total # of outcomes

Favorable outcomes = 10C1 as there are 10 pairs and we need ONE from these 10 pairs.
Total # of outcomes= 20C2 as there are 20 shoes and we are taking 2 from them.

P=10C1/20C2 =10/(19*10)=1/19

Answer: C.

tejal777 · Answer

Hey Bunuel thanks for the quick reply!However I am still not clear.Here is how our solution differs,
You are saying P= 10C1 /20C2 
While in my solution it is  20C1 /20C2

Can you please clarify this difference?Shoul'nt we be considering the answer wrt either pairs or total no. of shoes?in your answer in the numerator you are selecting one out of 10 PAIRS while I am selecting 1 out of 20 shoes..lol..is'nt it the same thing?(apparently not!)

jzd · Answer

I did it this way

1/10 * 1/19 * 10 = 1/19

Reasoning: The first shoe you pick is one of the pair out of 10 pairs so it is 1/10
The second shoe has to match the first shoe so it is now 1 out of 19 remain shoes (not pairs)
There are 10 pairs so the probably of the first 2 events times 10

I did not use any combination at all....

gmatjon · Answer

Bunuel, Your explanation is perfect!

lawschoolsearcher · Answer

My solution:

How many ways to select 2 from 20 shoes? 20!/2!18! = 190
How many ways to get matching shoes? 10 matching shoes = 10

10/190 = 1/19

anilisanil · Answer

just wondering what would be the answer if all 10 pairs were of same size.

In fact, I do not know why, but I inadvertently assumed that they are of same size and arrived at the answer 10/19, which I obviously did not find in the choices. Help please!

Bunuel · Answer

Size of the shoes is not the main point here. We are not told that these 10 pairs are identical (size, shape, color, ...) thus we cannot assume that.

Next, if we were told that there are 10 identical pairs of shoes, then the answer would be 10/19: after we choose one shoe, then there are 10 matching shoes left there out of 19, so P=10/19.

Hope it's clear.

EvaJager · Answer

You computed the probability of choosing a pair of shoes, one left and one right shoe, from 10 identical pairs. 
The probability to choose the first shoe is 20/20 = 1 (it can be any one of them). The probability to choose a matching shoe for the first one (meaning choose a left shoe after a right shoe, or the other way around) is 10/19.

In our case, when we have 10 different pairs of shoes, the probability to choose the first shoe is 20/20 = 1, then the probability to choose the matching shoe for the previous one is 1/19.

Chin926926 · Answer

My apologies if this is already somewhere on the forum, but I've searched and can't figure out any of the combination explanations because everyone always skips the last step. Can you please explain the math that went behind changing 10C1/20C2 into 1/19? I'm guessing this means you had to use the formula from the GMAT Book you all made?

So, I'm guessing this means...  \frac{10!}{1!9!}  divided by  \frac{20!}{2!18!}

If so, how did you quickly calculate the factorials like that? Is there a shortcut I'm missing?

Bunuel · Answer

10C1 = 10

20!/(2!18!) = 19*20/2 = 19*10.

10/(19*10) = 1/19.

Chin926926 · Answer

YOU SERIOUSLY JUST SAVED MY LIFE! THANK YOU!

EMPOWERgmatRichC · Answer

Hi All,

In these types of questions, you'll find that there are usually at least a couple of different ways that you can 'do the math' - however, there's the potential to make everything far more complex than it needs to be. Certain Quant questions are based heavily on logic, so if you can think through the logic involved, then the math itself will be rather minimal (and sometimes quite simple).

Here, we're dealing with 10 pairs of shoes and we're asked for the probability that selecting two random shoes will lead to a matching pair.

Since there are 20 shoes in total, you will end up choosing one of them with the first 'draw.' Whichever one you choose for that first shoe, it will have a match - 1 shoe out of the remaining 19 shoes will match the first one one that you 'drew', so the probability of selecting a matching pair is 1/19.

Final Answer:  C

GMAT assassins aren't born, they're made,
Rich

ScottTargetTestPrep · Answer

Solution:

The first shoe can be any shoe, so its probability is 1. However, there is only 1 out of 19 remaining shoes that can match the first shoe as a pair,so the probability that the second shoe matches is 1/19. Therefore, the probability of a matching pair is 1 x 1/19 = 1/19.

Answer: C

Riya1475828 · Answer

Here's my way of solving this problem:

Probability = No. of favorable outcome/Total no of outcome

Favorable outcome (Selecting matching pairs of shoes) = 20*1 (because there are 20 shoes to select from and 1 because each shoe will only have one other pair)
Total outcome (Selecting any shoe) = 20*19 (because there are 20 shoes to select from and 19 other shoes remaining for the second pick)

Probability of selecting matching pairs of shoes = 20/(20*19) = 1/19

Option C

Let me know if this helps!

colfer · Answer

First shoe can be selected in 20 ways. Only one way to get the second shoe and get a pair

Without restriction 20x19

Posted from my mobile device

Natansha · Answer

Hi Bunuel,

why can't we put denominator as 20C2 in the highlighted part, keeping numerator as same, since we are selecting 2 shoes out of 20?

Bunuel · Answer

The key here is maintaining consistency between the numerator and denominator. In the numerator, 20 represents the number of matching pairs where order matters, meaning we count both "Left → matching Right" and "Right → matching Left." Similarly, in the denominator, we count ordered pairs as 20 * 19, ensuring consistency.

Using 20C2 in the denominator would break this consistency because 20C2 represents unordered pairs of shoes, while the numerator still considers order. To maintain consistency, we could instead use 10 / 20C2, where both the numerator and denominator represent unordered pairs.

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A box contains 10 pairs of shoes (20 shoes in total). If two

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