Q. 3.25 - A First Course in Probability

Question

 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 ${\alpha }_1$ denote the proportion of time that a person under the age of $50$spends in the streets, and let${\alpha }_2$ be the corresponding value for those over$50$ . What quantity does the method suggested estimate? When is the estimate approximately equal to p ?

Step-by-Step Solution

Answer

The estimate approximately equal to p is click here.

Step1:Definition of conditional probability

The probability of any event A occurring when another event B in relation to A has already occurred is referred to as conditional probability. $P \left(A\right|B)$ represents it. As shown in the diagram above, S represents sample space, and A and B represent events.

Step2: Estimate value

p-proportion of population over age $50$ in town al-proportion of time that a person under the age of $50$ spends in the streets a2-proportion of time that a person over the age of $50$ spends in the streets This method estimate the value of $100,000p2a2/(al+a2$) This vLave estimated by this method will be close to$1000,000$ p if $al=A2$