Assuming that boys and girls are equally probable find the number of families out of 1600 families each having 5 children, with 3 boys
To find the number of families out of 1600 that have 5 children with 3 boys, we can use the concept of combinations.
First, let's determine the number of ways to choose 3 boys out of 5 children. This can be calculated using the combination formula:
C(n, r) = n! / (r! * (n-r)!)
Where n is the total number of children (5 in this case), and r is the number of boys (3 in this case).
Using the formula, we get:
C(5, 3) = 5! / (3! * (5-3)!)
= 5! / (3! * 2!)
= (5 * 4 * 3 * 2 * 1) / ((3 * 2 * 1) * (2 * 1))
= 10
So, there are 10 ways to choose 3 boys out of 5 children.
Now, since each child has an equal probability of being a boy or a girl, the probability of having 3 boys and 2 girls in a family is (1/2)^3 * (1/2)^2 = 1/32.
To find the number of families out of 1600 with 3 boys and 2 girls, we multiply the number of ways to choose 3 boys (10) by the probability of this combination (1/32):
Number of families = 10 * (1/32) = 10/32
Simplifying this fraction, we have:
Number of families = 5/16
Therefore, out of 1600 families, there would be 5/16 of them that have 3 boys and 2 girls among their 5 children.
To find the number of families out of 1600 families that each have 5 children, with 3 boys, we can use the concept of combinations.
First, let's determine the number of ways to choose 3 boys out of 5 children. This can be calculated using the combination formula:
C(n, r) = n! / (r! * (n - r)!)
In this case, n = 5 (total number of children in a family) and r = 3 (number of boys in a family).
C(5, 3) = 5! / (3! * (5 - 3)!)
= 5! / (3! * 2!)
= (5 * 4 * 3!) / (3! * 2)
= (5 * 4) / 2
= 10
So, there are 10 ways to have 3 boys out of 5 children in a family.
Now, we need to find the number of families that can have this combination. Since boys and girls are equally probable, we can say that there is a 50% chance of having a boy and a 50% chance of having a girl in each birth. Therefore, the probability of having 3 boys in a family can be calculated using the binomial probability formula:
P(x) = C(n, x) * p^x * (1 - p)^(n - x)
In this case, n = 5 (total number of children in a family), x = 3 (number of boys in a family), and p = 0.5 (probability of having a boy).
P(3) = C(5, 3) * (0.5)^3 * (1 - 0.5)^(5 - 3)
= 10 * (0.5)^3 * (0.5)^2
= 10 * 0.125 * 0.25
= 0.3125
So, the probability of having 3 boys out of 5 children in a family is 0.3125.
Finally, to find the number of families out of 1600 families with 3 boys, we multiply the probability by the total number of families:
Number of families = probability * total number of families
= 0.3125 * 1600
= 500
Therefore, there are 500 families out of 1600 families each having 5 children, with 3 boys.