Answer to Question #123478 in Statistics and Probability for Kingsley Owusu

Question

Suppose that a sample of 1000 household was initially selected and the respondents were asked whether they planned to purchase a large DASCO TV. Twelve months later the same respondents were asked whether they actually purchased the television. The results are summarized in the table below

Planned 

To purchase. Yes       No     Total

   Yes        200       50      250

   No        100       650     750

   Total       300       700    1000

i. Draw a tree diagram to represent the information in the table.

ii. Find the probability of selecting a respondent who actually purchased a large television. Interpret your answer in a simple single sentence.

iii. Compute the marginal probability of planning to purchase a large television.

iv. Find the probability of panned purchase or actual purchase.

v. Calculate the probability that a respondent actually purchased the large television given that he or she planned to purchase.

Answer 1

i. Let A denote the event that a respondent plan to purchase a large television.

Let B denote the event that a respondent actually purchased the large television.

$$\def\arraystretch{1.5} \begin{array}{c:c} & Purchase & No \ purchase & Total \\ \hline Plan & 200 & 50 & 250 \\ \hdashline No\ plan & 100 & 650 & 750 \\ \hdashline Total & 300 & 700 & 1000 \end{array}$$

ii.

Use the Law of Total Probability

$$P(B)=P(A)P(B|A)+P(A')P(B|A')=$$

$$=0.25(0.8)+(0.75)({2\over 15})=0.3$$

Or

$$P(B)={200+100\over 1000}=0.3$$

300 respondents from 1000 actually purchased the large television.

iii.

$$P(A)=0.25$$

iv.

$$P(A\cup B)={250+100\over 1000}=0.35$$

Or

$$P(A\cup B)=P(A)+P(B)-P(A\cap B)=$$

$$=0.25+0.3-{200\over 1000}=0.35$$

v.

$$P(B|A)={200\over 250}=0.8$$