Answer to Question #202132 in Statistics and Probability for Aasia Tariq

Question

Q.No.2 i. According to Nielsen Media Research, approximately 67% of all U.S. households with television have cable TV. Seventy-four percent of all U.S. households with television have two or more TV sets. Suppose 55% of all U.S. households with television have cable TV and two or more TV sets. A U.S. household with television is randomly selected.

a. What is the probability that the household has cable TV or two or more TV sets?

b. What is the probability that the household has cable TV or two or more TV sets but not both?

c. What is the probability that the household has neither cable TV nor two or more TV sets?

d. Why does the special law of addition not apply to this problem?

ii) a.  A batch of50 parts contains six defects. If two parts are drawn randomly one at a time without replacement, what is the probability that both parts are defective?

b. If this experiment is repeated, with replacement, what is the probability that both parts are defective?

Answer 1

Let C And T represent having cable and two television sets respectively.

$P(C)= 0.67$

$P(T)=0.74$

$P(C\cap T)=0.55$

a. $P(C\cup T)= P(C)+P(T)-P(C\cap T)$

$=0.67+0.74-0.55=0.86$

b. probability that the household has cable TV or two or more TV sets but not both is $P(C\cup T)-P(C\cap T)$

$=0.86-0.55=0.31$

c. $P(\bar C\cup\bar T)=1-P(C\cup T)$

$1-0.86= 0.14$

d. Special law of addition does not apply since the two events (having a cable tv and two or more TVs) are not mutually exclusive.

ii.

a. probability of two defectives is

$\frac{6}{50}+\frac{5}{49}=\frac{272}{1225}$

b. probability of two defectives is

$2(\frac{6}{50})=\frac{6}{25}$