Posted by **Genie** on Tuesday, October 2, 2012 at 7:43pm.

A drawer has 10 pairs of gloves. If I grab 5 gloves at random, what is the probability that I pick at least one matched pair? What is the probability that I pick at least one right glove and one left glove?

- Math -
**Reiny**, Tuesday, October 2, 2012 at 8:03pm
let's look at the prob that they are all different

start by picking any glove, now you have 1

there is 1 of the remaining 9 that will match

we don't want that, so the prob that the 2nd is NOT a match is 8/9

prob that the 2nd and third are NOT a match

= (8/9)(7/8)

prob that the 2nd, 3rd, 4th and 5th are NOT a match

= (8/9)(7/8)(6/7)(5/5)

so the prob that at least one match is found

= 1 - (8/9)(7/8)(6/7)(5/5)

= .....

Prob(at least one right and one left glove)

implies we don't want either all lefts or all rights

there are 5 lefts and 5 rights

prob(5 lefts) = (5/10)(4/9)(3/8)(2/7)(1/6)

the same would be true for prob(5 right)

so prob (all left or all right)

= (5/10)(4/9)(3/8)(2/7)(1/6) + (5/10)(4/9)(3/8)(2/7)(1/6)

= 2(5/10)(4/9)(3/8)(2/7)(1/6)

so prob(at least one left one right)

= 1 - 2(5/10)(4/9)(3/8)(2/7)(1/6)

= ...

- small typo - Math -
**Reiny**, Tuesday, October 2, 2012 at 8:08pm
in the first solution near the end

= (8/9)(7/8)(6/7)(5/5)

so the prob that at least one match is found

= 1 - (8/9)(7/8)(6/7)(5/5)

should have been:

= (8/9)(7/8)(6/7)(**5/6**)

so the prob that at least one match is found

= 1 - (8/9)(7/8)(6/7)(**5/6**)

- Math -
**Genie_math**, Wednesday, October 3, 2012 at 8:56pm
Wouldnt it be from 20 gloves as we have 10 pairs.

So for first glove we can pick any. [19 remaining]

For second glove we have to select from 18 which are different - 18/19

3rd glove - 16/18

4th glove - 14/17

5th glove - 12/16 ...

- Math -
**Reiny**, Wednesday, October 3, 2012 at 9:11pm
Of course you are right, how silly of me, there are obviously 20 gloves.

But.... why are you jumping from 18/19 to 16/18 etc

the pattern still continues following my argument above.

that is ...

(18/19)(17/18)(16/17) ....

- Math -
**Genie_math**, Wednesday, October 3, 2012 at 9:44pm
Ohk so it will be -

1st glove - 20

2nd glove - 18/19

3rd glove - 17/18

4th glove - 16/17

5th glove - 15/16

Total probability = 1 - the probability of above things right ??

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