Question

34​% of U.S. adults say they are more likely to make purchases during a sales tax holiday. You randomly select 10 adults. Find the probability that the number of adults who say they are more likely to make purchases during a sales tax holiday is​ (a) exactly​ two, (b) more than​ two, and​ (c) between two and​ five, inclusive.

Answer 1

Let $X=$ the number of adults who say they are more likely to make purchases during a sales tax holiday: $X\sim Bin(n, p).$

Given $n=10, p=0.34, q=1-p=0.66$

(a)

$$P(X=2)=\dbinom{10}{2}(0.34)^2(0.66)^{10-2}\approx0.187293$$

(b)

$$P(X>2)=1-P(X=0)-P(X=1)$$

$$-P(X=2)=1-\dbinom{10}{0}(0.34)^0(0.66)^{10-0}$$

$$-\dbinom{10}{1}(0.34)^1(0.66)^{10-1}-\dbinom{10}{2}(0.34)^2(0.66)^{10-2}$$

$$=1-0.01568336881-0.08079311205$$

$$-0.18729312338=0.71623039576$$

$$\approx0.716230$$

(c)

$$P(2\leq X\leq 5)=P(X=2)+P(X=3)$$

$$+P(X=4)+P(X=5)$$

$$=\dbinom{10}{2}(0.34)^2(0.66)^{10-2}$$

$$+\dbinom{10}{3}(0.34)^3(0.66)^{10-3}$$

$$+\dbinom{10}{4}(0.34)^4(0.66)^{10-4}$$

$$+\dbinom{10}{5}(0.34)^5(0.66)^{10-5}$$

$$\approx0.18729312+0.25729156$$

$$+0.23195224+0.14338866$$

$$\approx0.819926$$