魅影直播

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 ${伪}_1$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 ${伪}_2$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$P\left(B鈭�A^e\right)={伪}_1$.The probability that the person is found on the street given that the age of the person is more than 50 years is $P(B鈭�A)={伪}_2$

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

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

$=\left(P\right(A)脳P(B鈭�A\left)\right)+\left(P\left(A^c\right)脳P\left(B鈭�A^c\right)\right)$

$=p{伪}_2+(1-p){伪}_1$

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

$P(A鈭�B)=\frac{P(A鈭�B)}{P\left(B\right)}$

$=\frac{P(B鈭�A)脳P\left(A\right)}{P\left(B\right)}$

$=\frac{p{伪}_2}{p{伪}_2+(1-p){伪}_1}$

If ${伪}_1={伪}_2$, the value of$P(A鈭�B)$will be as follows:

$P(A鈭�B)=\frac{p{伪}_2}{p{伪}_2+(1-p){伪}_1}$

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

$=\frac{p{伪}_1}{{伪}_1}$

$=p$

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

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