Binomial Distribution (families and their children)

[Guest]

Four families have four children each. What is the probability that:

a) at least one of these families has two boys and two girls?

b) each family has at least one boy?

 

[tkhunny]

I think we need the probability of having a boy on a given birth.

 

[galactus]

Hints:

Assuming 50% chance of a boy and a 50% chance of a girl.

Think of the possible cases. If a family has 4 kids, they have the following possiblities:

BGGG
BBGG
BBBG
BBBB
GGGG

4 families have this possible procreation scenario.

At least one is the opposite of none.


You know, Pascal's triangle gives the probabilities of the different possible

combinations that can occur when an event is repeated a certain number

of times.

Look at the (P+1)st row of the triangle. In this case, the 5th row.

The coefficients are 1 4 6 4 1

The sum of these is 16

The probability of 4 boys is then 1/16
The probability of 3 boys and a girl is 4/16
The probability of 2 boys and 2 girls is 6/16\(\displaystyle \leftarrow\text{as per pka}\)
The probability of 1 boy and 3 girls is 4/16
The probability of 4 girls is 1/16

If you're curious.

 

[pka]

Four families have four children each. What is the probability that:a) at least one of these families has two boys and two girls?
b) each family has at least one boy?

It has been my experience that in human reproduction the 50% rule is used.
The probability that any family has two boys and two girls is 6/16 or 3/8.
Thus a) at least one of these families has two boys and two girls is 1-(5/8)<SUP>4</SUP>.

Part (b) is the most interesting of the questions: each family has at least one boy?
The probability of no boys is (1/2)<SUP>4</SUP>.
The probability of at least one boys is 1-(1/2)<SUP>4</SUP>.
The probability of each family has at least one boy is [1-(1/2)<SUP>4</SUP>]<SUP>4</SUP>.