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## Family Problem

 Category: Probability Submitted By: dalfamnest Fun: (2.31) Difficulty: (2.31)

The relatively unknown Mbwatzeze Tribe had a rather strange law restricting the size of families. Each married couple was expected to continue having children until they had EITHER one child of each sex OR a total of four children.

What was the average (mean) number of children born to each couple? (Assume that they kept to the law and that each couple had the maximum permitted by the law.)

2.75

Possible families with 2 children:
M-F or F-M
The probability of each of these is (0.5)^2, = 0.25.
So 0.25 + 0.25 = 0.5 had 2 children.

Possible 3-child families:
MMF or FFM
The probability of each of these is (0.5)^3, = 0.125.
So 0.125 + 0.125 = 0.25 had 3 children.

Possible 4-child families:
MMMM, MMMF, FFFF or FFFM
The probability of each of these is (0.5)^4, = 0.0625.
So 0.0625 x 4 = 0.25 had 4 children.

Half the couples had 2 children, a quarter had 3 and a quarter had 4. So the average (mean) size is
0.5 x 2 + 0.25 x 3 + 0.25 x 4 = 2.75