a class consists of 14 men and 16 women. A group of 5 is randomly chosen
a) the probability of this group containing at least 2 women and at least 2 men?
b) probability the group contains the same gender.
Total number of ways, N(all)
=30 choose 5
=C(30,5)
=30!/(5!(30-5)!)
A.
2 women+3 men
=16 choose 2 * 14 choose 3
=C(16,2)*C(14,3)
3 women+2 men
=16 choose 3 * 14 choose 2
=C(16,3)*C(14,2)
Number of ways to have at least two women and two men:
N1=C(16,2)*C(14,3)+C(16,3)*C(14,2)
Probability = N1/N(all)
B.
Number of ways for all men
=14 choose 5
=C(14,5)
Number of ways for all women
=16 choose 5
=C(16,5)
Number of ways for one gender
N2=C(14,5)+C(16,5)
Probability
=N2/N(all)
To calculate the probabilities, we need to determine the total number of possible groups of 5 people that can be formed from the class of 14 men and 16 women.
To calculate this, we can use the combination formula. The number of possible groups can be represented as:
C(30, 5)
where C(n, r) represents the binomial coefficient, also known as "n choose r", calculated as:
C(n, r) = n! / (r! * (n-r)!)
Now let's calculate the probabilities for each case:
a) Probability of the group containing at least 2 women and at least 2 men:
First, let's calculate the possible combinations where at least 2 women and at least 2 men are chosen:
1. Selecting exactly 2 women and 3 men:
C(16, 2) * C(14, 3)
2. Selecting exactly 3 women and 2 men:
C(16, 3) * C(14, 2)
3. Selecting exactly 4 women and 1 man:
C(16, 4) * C(14, 1)
4. Selecting exactly 5 women and 0 men:
C(16, 5) * C(14, 0)
Now, calculate the total number of possible combinations:
C(30, 5)
To find the probability, divide the sum of the combinations for the cases above by the total number of possible combinations:
P(at least 2 women and at least 2 men) = (1 + 2 + 3 + 4) / (C(30, 5))
b) Probability of the group containing the same gender:
First, let's calculate the possible combinations where either all men or all women are chosen:
1. Selecting 5 men:
C(14, 5)
2. Selecting 5 women:
C(16, 5)
Now, calculate the total number of possible combinations:
C(30, 5)
To find the probability, divide the sum of the combinations for the cases above by the total number of possible combinations:
P(same gender) = (1 + 2) / (C(30, 5))
Now, you can substitute the values into the formulas and calculate the probabilities.