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Birth Probability Puzzle
Difficulty Level

A family has seven children. Assume each of the children is equally likely to be either a boy or a girl.

1 ) If the children are listed in the order of birth (Example: B,B,G,B,G,G,B)
How many such listings are possible?

2 ) What is the probability that the family has exactly two girls?

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1 ) 128 Listing possibilities.
For each place in the order we have 2 chances (either boy or girl)
2x2x2x2x2x2x2 = 128 possibilities

2 ) 21 Combinations possibile for exact 2 girls.Lets take each scenario
Suppose first child is a Girl, So we have 6 possibilities in this case to have exactly 2 girls
G,X,X,X,X,X,X - Any of the 'X' can be a Girl child and we have 6 possibilities for that

G,G,X,X,X,X,X
G,X,G,X,X,X,X
G,X,X,G,X,X,X
G,X,X,X,G,X,X
G,X,X,X,X,G,X
G,X,X,X,X,X,G

Now we can take the case when second child is Girl: We have 5 possibilties
(Note first child will be a Boy in this case as we have considered first child- Girl possibiliry already
B,G,G,X,X,X,X
B,G,X,G,X,X,X
B,G,X,X,G,X,X
B,G,X,X,X,G,X
B,G,X,X,X,X,G

Possibilities when Third child is Girl: We have 4 possibilties
B,B,G,G,X,X,X
B,B,G,X,G,X,X
B,B,G,X,X,G,X
B,B,G,X,X,X,G

Possibilities when fourth child is Girl: We have 3 possibilties
B,B,B,G,G,X,X
B,B,B,G,X,G,X
B,B,B,G,X,X,G

Possibilities when Fifth child is Girl: We have 2 possibilties
B,B,B,B,G,G,X
B,B,B,B,G,X,G

Possibilities when Sixth child is Girl: We have only 1 possibilty
B,B,B,B,B,G,G

In short : 6+5+4+3+2+1 = 21

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