Probability - number of kids in a family (exactly 4 boys and 3 girls?)

dr.trovacek

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A family has 7 children. What is the probability that they have exactly 4 boys and 3 girls?
The solution given in the textbook: 0.27344

I've done problems where it is asked about the number of just one sex, for example exactly 2 boys. But I just can't crack this one. :confused:

Please help!
 
Solution

While I was waiting for the post to be approved I managed to get this solution with the help of
The Binomial distribution formula:

We look at the probability of having 3 girls then other "4 blank spaces" are boys. Event of family having a boy is a complementary event of an event that family has a girl, and their probabilities are both equal to \(\displaystyle \frac{1}{2}\) .



\(\displaystyle \binom{7}{3} (\frac{1}{2})^3 (\frac{1}{2})^4 = \binom{7}{3} (\frac{1}{2})^7 = 0.2734 \)
 
While I was waiting for the post to be approved I managed to get this solution with the help of
The Binomial distribution formula:

\(\displaystyle \binom{7}{3} (\frac{1}{2})^3 (\frac{1}{2})^4 = \binom{7}{3} (\frac{1}{2})^7 = 0.2734 \)

We look at the probability of having 3 girls and then other "4 blank spaces" are boys. Event of family having a boy is a complementary event of an event that family has a girl, and their probabilities are both equal to \(\displaystyle \frac{1}{2}\) .

Correct. In effect, you can restate the question with only one sex mentioned:

A family has 7 children. What is the probability that they have exactly 4 boys?

or

A family has 7 children. What is the probability that they have exactly 3 girls?

This, incidentally, shows why \(\displaystyle \binom{n}{r} = \binom{n}{n-r}\).
 
Correct. In effect, you can restate the question with only one sex mentioned:
A family has 7 children. What is the probability that they have exactly 4 boys?

or
A family has 7 children. What is the probability that they have exactly 3 girls?

This, incidentally, shows why \(\displaystyle \binom{n}{r} = \binom{n}{n-r}\).
I am always amused to see when a student has a two part question where part a is A family has 7 children. What is the probability that they have exactly 4 boys? and part b is A family has 7 children. What is the probability that they have exactly 3 girls? and after doing part a they do work for part b.
 
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