Question

the table below shows the results of a survey that asked 2871 people whether they are involved in any type of charity work. a person is selected at random from the sample. complete parts (a) through (d).

\

$$\begin{tabular}{ccccc} & frequently & occasionally & not at all & total \\\\ male & 225 & 452 & 793 & 1470 \\\\ female & 209 & 450 & 742 & 1401 \\\\ total & 434 & 902 & 1535 & 2871 \\end{tabular}$$

(a) find the probability that the person is frequently or occasionally involved in charity work.

p(being frequently involved or being occasionally involved) = 0.465
(round to the nearest thousandth as needed.)

(b) find the probability that the person is female or not involved in charity work at all.

p(being female or not being involved) = 0.764
(round to the nearest thousandth as needed.)

(c) find the probability that the person is male or frequently involved in charity work.

p(being male or being frequently involved) = 0.585
(round to the nearest thousandth as needed.)

(d) find the probability that the person is female or not frequently involved in charity work.

p(being female or not being frequently involved) =
(round to the nearest thousandth as needed.)

Explanation:

Identify the given values and target

Using the Two-Way Contingency Table Probability knowledge point, we extract the relevant frequencies from the contingency table:

• Total number of people surveyed, \(N = 2871\)
• Total number of females, \(n(\text{Female}) = 1401\)
• Total number of people frequently involved, \(n(\text{Frequently}) = 434\)
• Number of females who are frequently involved (the intersection), \(n(\text{Female} \cap \text{Frequently}) = 209\)

Our target is to find the probability that a randomly selected person is female or not frequently involved in charity work:

$$ P(\text{Female} \cup \text{Not Frequently}) $$

Apply the addition rule for probability

Using the Addition Rule for Probability knowledge point

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

Let \(A\) be the event that the person is female, and \(B\) be the event that the person is not frequently involved.
The number of people who are not frequently involved is:

$$ n(\text{Not Frequently}) = N - n(\text{Frequently}) = 2871 - 434 = 2437 $$

The number of people who are both female and not frequently involved is:

$$ n(\text{Female} \cap \text{Not Frequently}) = n(\text{Female}) - n(\text{Female} \cap \text{Frequently}) = 1401 - 209 = 1192 $$

Calculate the combined probability

Using the Addition Rule for Probability knowledge point

$$
LATEXBLOCK0
$$

Round to the nearest thousandth

Using the Addition Rule for Probability knowledge point

$$ P(\text{being female or not being frequently involved}) \approx 0.922 $$

Answer:

(d) Find the probability that the person is female or not frequently involved in charity work.

P(being female or not being frequently involved) = <blank>0.922</blank>
(Round to the nearest thousandth as needed.)