Question

short answer (8 points each). show your work for full credit.

1. the table below describes the smoking habits of a group of asthma sufferers.

\t\t\t\toccasional smoker\tregular smoker\theavy smoker\ttotal
\t\tnonsmoker\t\t\t\t\t\t\t\t
men\t342\t40\t69\t35\t486
women\t362\t44\t61\t37\t504
total\t704\t84\t130\t72\t990
if one of the 990 people is randomly selected,
(a) find the probability that the person is a woman.
(b) find the probability that the person is a man or a heavy smoker.
(c) find the probability that the person is a woman, given that the person is a regular smoker.
(d) find the probability that person is a woman and occasional smoker.
round all answers to 3 decimal places.

Explanation:

Step1: Recall Probability Formulas

For part (a): Probability of an event \( A \) is \( P(A)=\frac{n(A)}{n(S)} \), where \( n(A) \) is number of elements in \( A \), \( n(S) \) is total number of elements. Here, \( A \): person is a woman, \( n(A) = 504 \), \( n(S)=990 \).

For part (b): Use the addition rule \( P(A \cup B)=P(A)+P(B)-P(A \cap B) \), where \( A \): man, \( B \): heavy smoker. \( n(A)=486 \), \( n(B)=72 \), \( n(A \cap B)=35 \) (men who are heavy smokers).

For part (c): Conditional probability \( P(A|B)=\frac{P(A \cap B)}{P(B)}=\frac{n(A \cap B)}{n(B)} \), where \( A \): woman, \( B \): regular smoker. \( n(A \cap B)=61 \), \( n(B)=130 \) (total regular smokers).

For part (d): \( P(A \cap B)=\frac{n(A \cap B)}{n(S)} \), where \( A \): woman, \( B \): occasional smoker. \( n(A \cap B)=44 \), \( n(S)=990 \).

Step2: Calculate Part (a)

\( P(\text{woman})=\frac{504}{990}\approx0.509 \) (rounded to 3 decimals).

Step3: Calculate Part (b)

\( n(\text{man}) = 486 \), \( n(\text{heavy smoker}) = 72 \), \( n(\text{man and heavy smoker}) = 35 \)
\( P(\text{man or heavy smoker})=\frac{486 + 72 - 35}{990}=\frac{523}{990}\approx0.528 \)

Step4: Calculate Part (c)

\( n(\text{woman and regular smoker}) = 61 \), \( n(\text{regular smoker}) = 130 \)
\( P(\text{woman}|\text{regular smoker})=\frac{61}{130}\approx0.469 \)

Step5: Calculate Part (d)

\( n(\text{woman and occasional smoker}) = 44 \)
\( P(\text{woman and occasional smoker})=\frac{44}{990}\approx0.044 \)

Answer:

(a) \(\boldsymbol{0.509}\)
(b) \(\boldsymbol{0.528}\)
(c) \(\boldsymbol{0.469}\)
(d) \(\boldsymbol{0.044}\)