## Minimum Sample Size Needed to Include an Representative

Suppose in a village of 100 households there are just 3 households with no children. A sample is to be taken of all the households. How large must the sample size be before the probability that there is at least one childless household in the sample rises above 10%
Let
$x$
be the number of childless households in the sample, then we require
$P( x \ge 1) \ge 0.1$
or equivalently
$1-P(x \le 0) \ge 0.1 \rightarrow P(x \le 0)=P(x -0) \le 0.9$
.
If the sample size is 1 the probability that there are no childless households in the sample is
$\frac{97}{100}=0/97$
.
If the sample size is 2 the probability that there are no childless households in the sample is
$\frac{97}{100} \times \frac{96}{99}=0/9406$
.
If the sample size is 3 the probability that there are no childless households in the sample is
$\frac{97}{100} \times \frac{96}{99} \times \frac{95}{98}=0.9118$
.
If the sample size is 4 the probability that there are no childless households in the sample is
$\frac{97}{100} \times \frac{96}{99} \times \frac{95}{98} \times \frac{94}{97}=0.8836$
.
The minimum sample size is 4.