Answer to Question #106181 in Statistics and Probability for sadiaseher

Question

A mail-order firm has a circular that elicits a 10 percent response rate. Suppose 20 of the circulars are mailed as a market test in a new geographic area. Assuming that the 10 percent response rate is applicable in the new area, determine the probabilities of the following events:
(a) no one responds, (b) exactly two people respond, (c) a majority of the people respond, (d) less than 20 percent of the people respond.

Answer 1

Let $X=$ the number of people respond: $X\sim B(n;p)$

$$P(X=x)=\binom{n}{x}p^x(1-p)^{n-x}$$

Given that $p=0.1, n=20$

(a) The probability of no one responds is

$$P(X=0)=\binom{20}{0}0.1^0(1-0.1)^{20-0}=0.9^{20}\approx$$ $$\approx0.121577$$

(b) The probability of exactly two people respond is

$$P(X=2)=\binom{20}{2}0.1^2(1-0.1)^{20-2}=190(0.1)^2(0.9)^{18}\approx$$ $$\approx0.285180$$

(c) The probability of a majority of the people respond is

$$P(X>10)=1-P(X\leq10)=1-P(X=0)-$$ $$-P(X=1)-P(X=2)-P(X=3)-$$ $$-P(X=4)-P(X=5)-P(X=6)-$$ $$-P(X=7)-P(X=8)-P(X=9)-$$ $$-P(X=10)=1-\binom{20}{0}0.1^0(1-0.1)^{20-0}-$$

$$-\binom{20}{1}0.1^1(1-0.1)^{20-1}-\binom{20}{2}0.1^2(1-0.1)^{20-2}-$$

$$-\binom{20}{3}0.1^3(1-0.1)^{20-3}-\binom{20}{4}0.1^4(1-0.1)^{20-4}-$$

$$-\binom{20}{5}0.1^5(1-0.1)^{20-5}-\binom{20}{6}0.1^6(1-0.1)^{20-6}-$$

$$-\binom{20}{7}0.1^7(1-0.1)^{20-7}-\binom{20}{8}0.1^8(1-0.1)^{20-8}-$$

$$-\binom{20}{9}0.1^9(1-0.1)^{20-9}-\binom{20}{10}0.1^{10}(1-0.1)^{20-10}<$$

$$<0.000001\approx0$$

(d) The probability of less than 20 percent of the people respond is

$$20(0.2)=4$$

$$P(X<4)=P(X=0)+P(X=1)+$$ $$+P(X=2)+P(X=3)=$$

$$=\binom{20}{0}0.1^0(1-0.1)^{20-0}+\binom{20}{1}0.1^1(1-0.1)^{20-1}+$$

$$+\binom{20}{2}0.1^2(1-0.1)^{20-2}+\binom{20}{3}0.1^3(1-0.1)^{20-3}\approx$$

$$\approx0.867047$$