## Minimum Sample Size Needed to Include an Representative

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.