Answer to Question #127269 in Statistics and Probability for anaswar

Question

In certain town, 20% samples of the population are literate and assume that 200 investigators take samples of ten individuals to see whether they are literate. Find the probability that eight of the people are literate? How many investigators would you expect to report that 3 people or less are illiterates in the samples?

Answer 1

1. We shall denote by $X$ a random variable, that corresponds to the number of people that are literate among 10 selected individuals. It has a binomial distribution¹. The probability that eight people are literate is:

$P(X=8)=C_{10}^8(0.2)^8(0.8)^2=\frac{10\cdot9}{2}(0.2)^8(0.8)^2\approx0.0001$

(It is rounded to 4 decimal places)

2. At first we will look for the probability that one investigator reports that 3 people or less are illiterate. It has the same distribution as $X$. Namely, we have:

$P(X\leq3)=P(X=3)+P(X=2)+P(X=1)+P(X=0)=$

$=C_{10}^7(0.8)^3(0.2)^7+C_{10}^8(0.8)^2(0.2)^8+C_{10}^9(0.8)(0.2)^9+(0.2)^{10}\approx0.0009$

Now we denote by $Y$ a random variable that corresponds to the number of investigators that report that 3 people or less are illiterate. We suppose that all investigators make independent decisions. Then $Y$has a binomial distribution¹ with parameters n=200 and $p=0.0009$. We will look for such $k$ that $P(Y=k)$ has a maximum. Since $p$ is very small, we obtain:

$P(Y=0)=(0.9991)^{200}\approx0.8352$ (It is rounded to 4 decimal places)

Thus, we conclude that with the probability 0.8352 (83,52%) no investigators will report that 3 people or less are illiterate

References:

[1] Feller, W. (1967).  An Introduction to Probability Theory and Its Applications (Third ed.). New York: Wiley. p. 147

Answer: 1. 0.0001; 2. With the probability 0.8352 (83,52%) no investigators will report that 3 people or less are illiterate. (The numbers in answers are rounded to 4 decimal places)