Question

Random variable x represent the number of girls in a family of four children.
1) Construct a table describing the probability distribution, then find the mean and standard deviation. (Hint: List the different possible outcomes.)
2) Is it unusual for a family of four children to consist of four girls?

Answer 1

1) The possibilities are (B=Boy and G=Girl): BBBB, BBBG, BBGB, BBGG, BGBB, BGBG, BGGB, BGGG, GBBB, GBBG, GBGB, GBGG, GGB, GGBG, GGGB, GGGG Every outcome has an equal chance. Let X be the number of girls:
X - Probability
0 - 0.0625
1 - 0.25
2 - 0.375
3 - 0.25
4 - 0.0625
2) The meanis the sum of the product x and the probabilities of x:
\(\mu = \sum P(X) = 0 * 0.625 + 1 * 0.25 + 2 * 0.375 + 3 * 0.25 + 4 * 0.0625 = 2\)
The standard deviation is the square root of the sum of the squared deviations multiplied by the probability:
\(\sigma = \sqrt{\sigma(x-\mu)^{2}P(X)} = \sqrt{(0-2)^{2}*0.0625 +...+(4-2)^{2} *0.0625} = 2\)
It is unusual to get 4 girls, because 4 is standard deviations from the mean.
Result: \(\mu = 2,\ \sigma = 1,\ Unusual\)