Out of 800 families with 5 children each how many would you expect to have a-3 boys and 2 girls, b-either 2 or 3 boys, c-atleast one boy? Assume equal probability for boys and girls.
a) To calculate the number of families with 3 boys and 2 girls, we need to determine the probability of having 3 boys and 2 girls in each family.
The probability of having a boy is 1/2, and likewise for having a girl.
The probability of having 3 boys and 2 girls can be calculated using the binomial probability formula:
P(3 boys and 2 girls) = (5 choose 3) * (1/2)^3 * (1/2)^2
Using the formula for combinations (n choose k) = n! / (k! * (n-k)!)
P(3 boys and 2 girls) = (5 choose 3) * (1/2)^3 * (1/2)^2
P(3 boys and 2 girls) = 10 * (1/8) * (1/4)
P(3 boys and 2 girls) = 10/32
P(3 boys and 2 girls) = 5/16
Now, to find the number of families, we multiply the probability by the total number of families:
Number of families with 3 boys and 2 girls = (5/16) * 800
Number of families with 3 boys and 2 girls = 250
So, out of 800 families, we would expect 250 to have 3 boys and 2 girls.
b) To find the number of families with either 2 or 3 boys, we need to calculate the probability of having 2 or 3 boys in each family and then multiply it by the total number of families.
The probability of having 2 boys can be calculated using the binomial probability formula:
P(2 boys) = (5 choose 2) * (1/2)^2 * (1/2)^3
Using the formula for combinations (n choose k):
P(2 boys) = (5 choose 2) * (1/2)^2 * (1/2)^3
P(2 boys) = 10 * (1/4) * (1/8)
P(2 boys) = 10/32
P(2 boys) = 5/16
Similarly, the probability of having 3 boys can be calculated:
P(3 boys) = (5 choose 3) * (1/2)^3 * (1/2)^2
P(3 boys) = 10 * (1/8) * (1/4)
P(3 boys) = 10/32
P(3 boys) = 5/16
Therefore, the probability of having either 2 or 3 boys is the sum of these probabilities:
P(either 2 or 3 boys) = P(2 boys) + P(3 boys)
P(either 2 or 3 boys) = 5/16 + 5/16 = 10/16
Now, we find the number of families with either 2 or 3 boys:
Number of families with either 2 or 3 boys = (10/16) * 800
Number of families with either 2 or 3 boys = 500
So, out of 800 families, we would expect 500 to have either 2 or 3 boys.
c) To find the number of families with at least one boy, we can calculate the probability of having no boys and subtract it from 1.
The probability of having no boys can be calculated using the binomial probability formula:
P(no boys) = (5 choose 0) * (1/2)^0 * (1/2)^5
Using the formula for combinations (n choose k):
P(no boys) = (5 choose 0) * (1/1) * (1/32)
P(no boys) = 1 * 1/32
P(no boys) = 1/32
Therefore, the probability of having at least one boy is:
P(at least one boy) = 1 - P(no boys)
P(at least one boy) = 1 - 1/32
P(at least one boy) = 31/32
Now, we find the number of families with at least one boy:
Number of families with at least one boy = (31/32) * 800
Number of families with at least one boy ≈ 775
So, out of 800 families, we would expect approximately 775 to have at least one boy.