PROBLEM 6:
Roberto has a collection of 10 pairs of gloves in his wardrobe. Before a business trip, he has to pack his luggage, and he selects 8 gloves, without looking at them. We assume that any set of 8 gloves is equally likely to be chosen. Find the probability that these 8 gloves do not include any matching pair of gloves, that is, that there are no two (left and right) gloves, coming from the same pair.
(Enter an exact answer or an answer accurate to at least 3 decimal places.)
probability of success for each choice:
1st glove: 20/20
2nd glove: 18/19
3rd glove: 16/18
...
8th glove: 4/13
so multiply all of those together: 384/4199 ≈ 0.09145
the answer is 0.09145
The last one is 6/13
To find the probability that the 8 selected gloves do not include any matching pair, we need to consider the number of ways this can happen and divide it by the total number of possible outcomes.
First, let's calculate the total number of ways to select 8 gloves out of the 10 pairs available.
We have 10 pairs of gloves, which means there are 20 gloves in total. To select 8 gloves, we need to choose 8 out of the 20 available gloves. This can be done using combinations.
The number of ways to select 8 gloves out of 20 is denoted as C(20, 8) and can be calculated using the formula for combinations:
C(n, r) = n! / (r! * (n - r)!)
Plugging in the values, we have:
C(20, 8) = 20! / (8! * (20 - 8)!)
Now, let's determine the number of ways to select 8 gloves such that there are no matching pairs. This means each of the 8 gloves chosen should be from a different pair.
Out of the 10 pairs of gloves, we need to select 8 different pairs. This can be calculated using combinations, and the formula becomes:
C(10, 8) = 10! / (8! * (10 - 8)!)
Next, for each pair selected, we have 2 gloves to choose from since we need both the left and right glove. So, we have 2 ways to choose from each of the 8 selected pairs.
Therefore, the total number of ways to select 8 gloves without any matching pair is:
C(10, 8) * (2^8)
Finally, we can find the probability by dividing the number of successful outcomes (8 gloves without any matching pair) by the total number of possible outcomes (8 gloves selected from 10 pairs):
P = (C(10, 8) * (2^8)) / C(20, 8)
Calculating these values, we get:
C(10, 8) = 10! / (8! * (10 - 8)!) = 45
C(20, 8) = 20! / (8! * (20 - 8)!) = 125,970
P = (45 * (2^8)) / 125,970 = 0.108
Therefore, the probability that the 8 selected gloves do not include any matching pair is approximately 0.108, accurate to three decimal places.