The probability of a child being a boy instead of a girl is 50% or 1/2. If a couple has 6 children, what is the probabilty that they had 3 boys and 3 girls?

Question

Please express the answer in either percent, fraction, or ratio form. Thank you in advance! 

Answer 1

First, calculate the numerator of the probability, the number of ways to have exactly 3 boys among the 6 children.

 

There are 6 children: 1st-born, 2nd-born, 3rd-born, 4th-born, 5th-born, 6th-born.

 

We want the number of ways we can choose 3 out of these 6 for the 3 boys.  This is "6 choose 3" or 6C3 = 20.

 

Then, we calculate the denominator of the probability, the number of ways that any of the 6 can have either sex.

 

There are 2 choices for the 1st-born (B or G), then

there are 2 choices for the 2nd-born (B or G), then

there are 2 choices for the 3rd-born (B or G), then

there are 2 choices for the 4th-born (B or G), then

there are 2 choices for the 5th-born (B or G), then

there are 2 choices for the 6th-born (B or G).

 

That's 2*2*2*2*2*2 = 2⁶ = 64

 

  So the desired probability is 20/64 or 5/16.

Answer 2

(1/2)⁶=1/64