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# a family has 5 children, what is the probability that there

tagged by: GMATinsight

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A

B

C

D

E

## Global Stats

Difficult

a family has 5 children, what is the probability that there are 3 boys and 2 girls among the children?

A) 1/32
B) 1/16
C) 3/32
D) 1/4
E) 5/16

Source: www.GMATinsight.com

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The problem should make clear that P(boy) = P(girl) = 1/2:

Quote:
A family has 5 children. If each child is equally likely to be a boy or a girl, what is the probability that there are 3 boys and 2 girls among the children?

A) 1/32
B) 1/16
C) 3/32
D) 1/4
E) 5/16
P(exactly n times) = P(one way) * total possible ways.

Let B = boy and G = girl.

P(one way):
One way to get exactly 3 boys and 2 girls is BBBGG.
P(BBBGG) = 1/2 * 1/2 * 1/2 * 1/2 * 1/2 = 1/32.

Total possible ways:
Any arrangement of the letters BBBGG will yield exactly 3 boys and 2 girls.
Thus, to account for all the ways to get exactly 3 boys and 2 girls, the result above needs to be multiplied by the number of ways to arrange BBBGG.
Number of ways to arrange BBBGG = 5!/(3!2!) = 10.

Multiplying the results above, we get:
P(exactly 3 boys and 2 girls) = 10 * 1/32 = 10/32 = 5/16.

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### GMAT/MBA Expert

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GMATinsight wrote:
A family has 5 children, what is the probability that there are 3 boys and 2 girls among the children?

A) 1/32
B) 1/16
C) 3/32
D) 1/4
E) 5/16

Source: www.GMATinsight.com
$? = P\left( {3B\,\,{\text{and}}\,\,2G\,\,{\text{among}}\,\,5\,\,{\text{children}}} \right)$

Imagine 3 B´s and 2 G´s in a row... each sequence is associated with the children´s sexuality, from (say) the older to the younger child.

Total number: 2^5 = 32 equiprobable sequences (for instance starting with BBBBB and ending with GGGGG.)

Favorable number: C(5, 3) = 10 , because we must choose among the 5 positions in the sequence, 3 of them to "put" the B´s (and the remaining 2 to "put" the G´s).

$? = \frac{{C\left( {5,3} \right)}}{{32}} = \frac{{10}}{{32}} = \frac{5}{{16}}$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
fskilnik.

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### GMAT/MBA Expert

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GMATinsight wrote:
a family has 5 children, what is the probability that there are 3 boys and 2 girls among the children?

A) 1/32
B) 1/16
C) 3/32
D) 1/4
E) 5/16
The number of 5 children of any sex is 2 x 2 x 2 x 2 x 2 = 2^5 = 32. The number of ways that the 5 children could be 3 boys and 2 girls is 5!/(3! x 2!) = (5 x 4)/2! = 10. Therefore, the probability is 10/32 = 5/16.

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