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
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:

Part (a)

Step 1: Identify relevant counts

The total number of people is \( N = 990 \). The number of women is \( n(\text{women}) = 504 \).

Step 2: Apply probability formula

The probability \( P(\text{woman})=\frac{n(\text{women})}{N}=\frac{504}{990}\approx0.509 \)

Part (b)

Step 1: Identify counts for man or heavy smoker

Number of men: \( n(\text{men}) = 486 \)
Number of heavy smokers: \( n(\text{heavy smoker}) = 72 \)
Number of men who are heavy smokers: \( n(\text{men and heavy smoker}) = 35 \)
Using the formula \( P(A\cup B)=P(A)+P(B)-P(A\cap B) \), where \( A \) is "man" and \( B \) is "heavy smoker".

Step 2: Calculate each probability

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

Part (c)

Step 1: Identify counts for regular smoker and woman regular smoker

Number of regular smokers: \( n(\text{regular smoker}) = 130 \)
Number of women regular smokers: \( n(\text{woman and regular smoker}) = 61 \)

Step 2: Apply conditional probability formula

\( P(\text{woman}|\text{regular smoker})=\frac{P(\text{woman and regular smoker})}{P(\text{regular smoker})}=\frac{\frac{61}{990}}{\frac{130}{990}}=\frac{61}{130}\approx0.469 \)

Part (d)

Step 1: Identify count for woman and occasional smoker

Number of women who are occasional smokers: \( n(\text{woman and occasional smoker}) = 44 \)

Step 2: Apply probability formula

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

Answer:

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