There are 5 black mice and 5 white mice available for an experiment. A random selection is made from the set of ten mice. What is the probability that two black mice and one white mouse will be selected? Thanks
To find the probability of selecting two black mice and one white mouse, we need to first determine the total number of possible outcomes.
Since there are 10 mice in total (5 black and 5 white), the total number of possible outcomes is given by selecting 3 mice at random from the set of 10 mice.
To calculate this, we can use the concept of combinations. The number of ways to select 3 mice from a set of 10 mice is denoted as "10 choose 3" and can be calculated as:
10! / (3!(10-3)!)
= 10! / (3!7!)
where "!" denotes the factorial function.
Calculating the factorial values, we have:
10! = 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = 3,628,800
3! = 3 * 2 * 1 = 6
7! = 7 * 6 * 5 * 4 * 3 * 2 * 1 = 5,040
Substituting these values into the formula:
= 3,628,800 / (6 * 5,040)
= 3,628,800 / 30,240
= 120
Therefore, there are 120 possible outcomes for selecting 3 mice from the set of 10 mice.
Next, we need to determine the number of ways to select 2 black mice and 1 white mouse.
To calculate this, we can use the concept of combinations again. The number of ways to select 2 black mice from the available 5 black mice can be calculated as "5 choose 2". Similarly, the number of ways to select 1 white mouse from the available 5 white mice can be calculated as "5 choose 1".
Using the combination formula, we have:
"5 choose 2" = 5! / (2!(5-2)!) = (5 * 4) / (2 * 1) = 10
"5 choose 1" = 5! / (1!(5-1)!) = 5 / 1 = 5
Therefore, there are 10 ways to select 2 black mice and 5 ways to select 1 white mouse.
To find the probability, we divide the number of favorable outcomes by the total number of possible outcomes:
Probability = (number of favorable outcomes) / (total number of possible outcomes)
= (10 * 5) / 120
= 50 / 120
= 5/12
So, the probability of selecting two black mice and one white mouse is 5/12 or approximately 0.4167.
To find the probability of selecting two black mice and one white mouse, we need to calculate the number of favorable outcomes and divide it by the total number of possible outcomes.
First, let's find the total number of possible outcomes:
There are 10 mice in total, and we are selecting 3 of them. This can be calculated using the combination formula: C(n, r) = n! / (r! * (n - r)!).
So, the total number of possible outcomes is: C(10, 3) = 10! / (3! * (10 - 3)!) = 10! / (3! * 7!) = 120 / (6 * 5040) = 120 / 5040 = 1 / 42.
Next, let's find the number of favorable outcomes:
We need to select 2 black mice and 1 white mouse.
There are 5 black mice, and we need to choose 2 of them. This can be calculated using the combination formula: C(5, 2) = 5! / (2! * (5 - 2)!) = 120 / (2 * 6) = 10.
There are 5 white mice, and we need to choose 1 of them. This can be calculated using the combination formula: C(5, 1) = 5! / (1! * (5 - 1)!) = 120 / (1 * 24) = 5.
So, the number of favorable outcomes is: C(5, 2) * C(5, 1) = 10 * 5 = 50.
Now, we can calculate the probability:
Probability = Number of favorable outcomes / Total number of possible outcomes = 50 / (1 / 42) = 50 * 42 = 2100.
Therefore, the probability that two black mice and one white mouse will be selected is 2100.