The following method was proposed to estimate the number of people over the age of 50 who reside in a town of known population 100,000: “As you walk along the streets, keep a running count of the percentage of people you encounter who are over 50. Do this for a few days; then multiply the percentage you obtain by 100,000 to obtain the estimate.” Comment on this method. Hint: Let p denote the proportion of people in the town who are over 50. Furthermore, let α1 denote the proportion of time that a person under the age of 50 spends in the streets, and let α2 be the corresponding value for those over 50. What quantity does the method suggest estimate? When is the estimate approximately equal to p?

In this question, given that the method of estimating the number of people whose age is over 50 in a known population is provided. Also given the number of people in the population is 100000.

Consider an event $A$that represents the age of the person is more than 50 years and $B$is an event which represents that the person is found in the street. Moreover, $p$denotes the proportion of people in the town who are over 50 and ${\alpha }_1$denote the proportion of time that a person under the age of 50 spends in the streets. Therefore, it can be concluded that$1-p$denotes the proportion of people in the town who are under 50 and ${\alpha }_2$denotes the proportion of time that a person over the age of 50 spends in the street.

The information can change in terms of probability. The probability that the age of the person is more than 50 years is $P\left(A\right)=p$and the probability that the age of the person is less than 50 years is$P\left(A^c\right)=1-p$. The probability that the person is found on the street given that the age of the person is less than 50 years is$P\left(B\mid A^e\right)={\alpha }_1$.The probability that the person is found on the street given that the age of the person is more than 50 years is $P(B\mid A)={\alpha }_2$

The probability that the person is found on the street is calculated as follows:

$P\left(B\right)=P(B\cap A)+P\left(B\cap A^c\right)$

$=\left(P\right(A)\times P(B\mid A\left)\right)+\left(P\left(A^c\right)\times P\left(B\mid A^c\right)\right)$

$=p{\alpha }_2+(1-p){\alpha }_1$

The probability that the age of the people is more than 50 years given that people is on the street is calculated as follows:

$P(A\mid B)=\frac{P(A\cap B)}{P\left(B\right)}$

$=\frac{P(B\mid A)\times P\left(A\right)}{P\left(B\right)}$

$=\frac{p{\alpha }_2}{p{\alpha }_2+(1-p){\alpha }_1}$

If ${\alpha }_1={\alpha }_2$, the value of$P(A\mid B)$will be as follows:

$P(A\mid B)=\frac{p{\alpha }_2}{p{\alpha }_2+(1-p){\alpha }_1}$

$=\frac{p{\alpha }_1}{p{\alpha }_1+(1-p){\alpha }_1}$ $\text{If}{\alpha }_1={\alpha }_2$

$=\frac{p{\alpha }_1}{{\alpha }_1}$

$=p$

The suggested method estimates the desired quality, if people on street can be over 50 or fewer than 50 with equal probability. That is${\alpha }_1={\alpha }_2$

Therefore, the provided method estimates the proportion of the in the town who are over 50 if${\alpha }_1={\alpha }_2$