# Math Help - probaility Question

1. ## probability Question

Please any one explain me,as soon as possible how to get answer for below question.Any help really appreciated.Thank you.

A family of five children is known to have at least two girls. What is the probability of this family having exactly four girls?

a. 5/30
b. 5/26
c. 8/10
d. 5/10
e. 11/30

2. It is not impossible to list all possiblities. Since each child can be either boy or girl there are $2^5= 32$ possibilities: BBBBB, BBBBG, BBBGB, etc.

Rather than do that, I will note that exactly one has no girls: BBBBB. Five have one girl: GBBBB, BGBBB, BBGBB, BBBGB, and BBBBG. That leaves 32- 6= 26 that have two or more girls. Therefore, your answer will be a fraction having 26 as denominator, not 32.

It should be easy to see that there are also exactly five ways to list "four girls": BGGGG, GBGGG, GGBGG, GGGBG, and GGGGB.

3. Originally Posted by lasantha
Please any one explain me,as soon as possible how to get answer for below question.Any help really appreciated.Thank you.

A family of five children is known to have at least two girls. What is the probability of this family having exactly four girls?

a. 5/30
b. 5/26
c. 8/10
d. 5/10
e. 11/30
Here is an alternative way.

The only possibilities are... 2 girls and 3 boys, 3 girls and 2 boys, 4 girls and 1 boy, 5 girls

Therefore, the probability of exactly 4 girls is

$\frac{number\ of\ ways\ to\ have\ 4\ girls}{number\ of\ ways\ to\ have\ 2,\ 3,\ 4\ or\ 5\ girls}$

$=\displaystyle\huge\frac{\binom{5}{4}}{\binom{5}{2 }+\binom{5}{3}+\binom{5}{4}+\binom{5}{5}}$

$=\frac{5}{10+10+5+1}$