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?
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)
= ...
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)
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 ...
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) ....
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 ??
To find the probability of picking at least one matched pair of gloves, we need to consider the different scenarios in which this can happen.
First, let's determine the total number of possible outcomes when picking 5 gloves out of 10 pairs. This can be calculated using combinations. Since there are 10 pairs, there are 20 gloves in total. Therefore, the number of ways to choose 5 gloves out of 20 is given by the combination formula:
C(20, 5) = 20! / (5! * (20 - 5)!)
Now, let's consider the different scenarios in which we pick at least one matched pair of gloves.
1. Picking a matched pair of gloves (both gloves from the same pair):
- There are 10 pairs to choose from.
- Once we choose a pair, we need to select one glove from it.
- The remaining 3 gloves can be chosen from the remaining 18 gloves.
Hence, the number of ways to pick at least one matched pair is:
10 * 2 * C(18, 3)
2. Picking two matched pairs of gloves:
- There are 10 pairs to choose the first matched pair from.
- Once we choose the first pair, we need to select one glove from it.
- There are 9 remaining pairs, from which we can choose the second matched pair.
- Once we pick the second pair, we need to select one glove from it.
- The remaining glove can be chosen from the remaining 16 gloves.
Hence, the number of ways to pick at least two matched pairs is:
10 * 2 * 9 * 2 * C(16, 1)
To calculate the probability, we divide the sum of these two scenarios by the total number of possible outcomes:
Probability of picking at least one matched pair = (10 * 2 * C(18, 3) + 10 * 2 * 9 * 2 * C(16, 1)) / C(20, 5)
Now, let's move on to the second part of the question: the probability of picking at least one right glove and one left glove.
To solve this, we can consider two scenarios:
1. Picking a right glove and a left glove (one glove from each pair):
- There are 10 pairs, so we can choose a pair in 10 ways.
- Once we choose a pair, we need to select one glove from it.
- There are 9 remaining pairs, so we can select another pair in 9 ways.
- Once we pick the second pair, we need to select one glove from it.
- The remaining 3 gloves can be chosen from the remaining 18 gloves.
Hence, the number of ways to pick at least one right and one left glove is:
10 * 2 * 9 * 2 * C(18, 3)
2. Picking two right gloves from two different pairs or two left gloves from two different pairs:
- There are 10 pairs, so we can choose a pair in 10 ways.
- Once we choose a pair, we need to select one glove from it.
- There are 9 remaining pairs, so we can select another pair in 9 ways.
- Once we pick the second pair, we need to select one glove from it.
- The remaining glove can be chosen from the remaining 18 gloves.
Hence, the number of ways to pick two right gloves or two left gloves is:
10 * 2 * 9 * 2 * C(18, 4)
To calculate the probability, we add up the two scenarios and divide by the total number of possible outcomes:
Probability of picking at least one right glove and one left glove = (10 * 2 * 9 * 2 * C(18, 3) + 10 * 2 * 9 * 2 * C(18, 4)) / C(20, 5)