A committee of two persons is selected from two men and two women.
Find the probability that the committee will have at least one man.
find the cumulative function, mean and variance of the following continuous pdf of a random variable X. f(x)=x/2 for 0<x<1 f(x)= 1/2 for 1<x<=2 f(x)=(3-x)/2 for 2<x<3 f(x)=0 otherwise
find the cumulative function, mean and variance of the following continuous pdf of a random variable X. f(x)=x/2 for 0<x<1 f(x)= 1/2 for 1<x<=2 f(x)=(3-x)/2 for 2<x<3 f(x)=0 otherwise
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To find the probability that the committee will have at least one man, we can calculate the probability of the committee having no men and subtract it from 1.
Step 1: Find the total number of possible committees
The total number of possible committees can be calculated using the combination formula. Since we need to select two people from a total of four, we use the formula:
C(n, r) = n! / (r! * (n-r)!),
where C is the combination, n is the total number of items, and r is the number of items to be selected.
In this case, n = 4 (total number of people) and r = 2 (committee size), so the total number of possible committees is:
C(4, 2) = 4! / (2! * (4-2)!) = 6.
Step 2: Find the number of committees with no men
If the committee is to have no men, we need to select both members from the two women. So, the number of committees with no men is:
C(2, 2) = 2! / (2! * (2-2)!) = 1.
Step 3: Calculate the probability of having at least one man
The probability of having at least one man is equal to 1 minus the probability of having no men. So,
Probability of having at least one man = 1 - (Number of committees with no men / Total number of possible committees)
Plugging in the values, we get:
Probability of having at least one man = 1 - (1 / 6) = 5/6.
Therefore, the probability that the committee will have at least one man is 5/6.
To find the probability that the committee will have at least one man, we first need to find the total number of possible committees and then the number of committees with at least one man.
Step 1: Find the total number of possible committees
In this case, we have 2 men and 2 women to choose from. To find the total number of possible committees, we can use the combination formula.
The number of ways to select a committee of two persons from a group of four is given by:
C(4, 2) = 4! / (2! * (4-2)!) = 6
Hence, there are a total of 6 possible committees.
Step 2: Find the number of committees with at least one man
To find the number of committees with at least one man, we need to consider two scenarios: a committee with exactly one man and a committee with two men.
Scenario 1: Committee with exactly one man
We have 2 men to choose from and we need to select 1 man, so we can use the combination formula again.
The number of ways to select a committee with one man from a group of two is given by:
C(2, 1) = 2
Scenario 2: Committee with two men
Since there are only 2 men available, there is only one way to select a committee with both men.
So, the total number of committees with at least one man is:
2 (from scenario 1) + 1 (from scenario 2) = 3
Step 3: Calculate the probability
The probability of an event occurring is the number of favorable outcomes divided by the total number of possible outcomes.
Therefore, the probability that the committee will have at least one man is:
P(committee with at least one man) = number of committees with at least one man / total number of possible committees
P(committee with at least one man) = 3 / 6 = 1/2 = 0.50
Hence, the probability is 0.50 or 50%.