### Brain Teasers

# Family Problem

Probability
Probability puzzles require you to weigh all the possibilities and pick the most likely outcome.Probability

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.)

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.)

### Hint

Start by listing the possible families and determine the probability of each.e.g. Male-Male-Female-STOP. P(MMF) = (0.5)^3 = 0.125

A probability tree will help, for those who know how to construct one.

### Answer

2.75Possible 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

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