Q. 8.17 - A First Course in Probability

Question

Redo Example$5b$ under the assumption that the number of man-woman pairs is (approximately) normally distributed. Does this seem like a reasonable supposition?

Step-by-Step Solution

Answer

Therefore,

$P\{X\le 30\}\approx 0$.

Step 1 Given Information.

the number of man-woman pairs is (approximately) normally distributed.

Step 2 Explanation.

Consider Example$5 b$(textbook). Now, assume that the number of man-woman pairs

$X=\sum _{i=1}^{100}{X}_i,\text{ where }{X}_i=\left\{\begin{array}{l}1,\text{ man }i\text{ is paired with a woman }\\ 0,\text{ otherwise }\end{array}\right.$

is approximately normally distributed with a mean$E\left[X\right]\approx 50.25$ and a variance $Var\left(X\right)\approx 25.126$

The probability that most $30$pairs will consist of a man and a woman is$P\{X\le 30\}$. To approximate this probability we can use the central limit theorem and in that case, we get:

$P\{X\le 30\}$$\approx P\left\{\frac{X-E\left[X\right]}{\sqrt{Var\left(X\right)}}\le \frac{30-E\left[X\right]}{\sqrt{Var\left(X\right)}}\right\}$$=P\left\{\frac{X-50.25}{\sqrt{25.126}}\le \frac{30-50.25}{\sqrt{25.126}}\right\}$

$\approx 0$ .