Question

3. On a large college campus, 84% of the students report drinking alcohol within the past month, 33% report using some type of tobacco product within the past month, and 86% report using at least one of these two substances during the past month. In an experiment, whether or not a randomly selected student has drunk alcohol or used a tobacco product during the past month is recorded. Using set notation, symbolically denote each of the following events and then determine the proportion of the student body represented by each a The selected student has both drunk alcohol and used a tobacco product during the past month. b The selected student has abstained from both alcohol and tobacco products during the past month. c The selected student has either consumed alcohol or used a tobacco product during the past month but not both.

Answer 1

Proportion of students drinking alcohol, P(A) = 0.84

Proportion of students using Tobacco, P(T) = 0.33

Proportion of students ents using at least one of the above substances, P(AUT) = 0.86

A.

Proportion of students has used both substances = $\fn_phv P(A\cap T)$

$\fn_phv P(A\cup T) =P(A) + P(T) - P(A\cap T)$

0.86 = 0.84 + 0.33 - $\fn_phv P(A\cap T)$

$\fn_phv P(A\cap T)$ = 0.84 + 0.33 - 0.86

$\fn_phv P(A\cap T)$ = 0.31

B.

Proportion of students that has abstained from both substances = $\fn_phv P(A'\cap B')$

$\fn_phv P(A'\cap B') = P(A\cup B)' = 1-P(A\cup B)$

$\fn_phv P(A'\cap B') =$ 1-0.86

$\fn_phv P(A'\cap B') =$ 0.14

C.

Proportion of students who have used either of one substance but not both = $\fn_phv P(A\Delta T)$

$\fn_phv P(A\Delta T) = P(A\cup T)-P(A\cap T)$

$\fn_phv P(A\Delta T)$ = 0.86 - 0.31

$\fn_phv P(A\Delta T)$ = 0.55