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) =
(round to the nearest thousandth as needed.)

Explanation:

Identify the given values and target

Using the Two-Way Contingency Table Probability knowledge point

• Total sample size \(N = 2871\)
• Total number of males \(n(\text{Male}) = 1470\)
• Total number of people frequently involved \(n(\text{Frequently}) = 434\)
• Number of males who are frequently involved \(n(\text{Male} \cap \text{Frequently}) = 225\)
• Target: Find \(P(\text{being male or being frequently involved})\)

Apply the addition rule for probability

Using the Addition Rule for Probability knowledge point

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

$$ P(\text{Male} \cup \text{Frequently}) = \frac{n(\text{Male}) + n(\text{Frequently}) - n(\text{Male} \cap \text{Frequently})}{N} $$

Calculate the combined frequency

Using the Addition Rule for Probability knowledge point

$$ n(\text{Male} \cup \text{Frequently}) = 1470 + 434 - 225 = 1679 $$

Compute and round the probability

Using the Two-Way Contingency Table Probability knowledge point

$$ P(\text{Male} \cup \text{Frequently}) = \frac{1679}{2871} \approx 0.5848136 $$

Rounding to the nearest thousandth gives \(0.585\).

Answer:

Find the probability that the person is male or frequently involved in charity work.

P(being male or being frequently involved) = <blank>0.585</blank>
(Round to the nearest thousandth as needed.)