Questions: The table below shows the results of a survey that asked 2866 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). Frequently Occasionally Not at all Total ---------------------------------------------------- Male 227 459 799 1485 Female 205 430 746 1381 Total 432 889 1545 2866 (a) Find the probability that the person is frequently or occasionally involved in charity work. P (being frequently involved or being occasionally involved) = 0.461 (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) = 0.761 (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) = 0.590 (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.)

Solution Steps

To solve these probability questions, we will use the principle of inclusion-exclusion. For each part, we will calculate the probability of each individual event and then adjust for any overlap between the events.

(a) To find the probability that a person is frequently or occasionally involved in charity work, we add the probabilities of being frequently involved and occasionally involved.

(b) To find the probability that a person is female or not involved in charity work at all, we add the probabilities of being female and not involved, and subtract the probability of being both female and not involved.

(c) To find the probability that a person is male or frequently involved in charity work, we add the probabilities of being male and frequently involved, and subtract the probability of being both male and frequently involved.

To find the probability that a person is frequently or occasionally involved in charity work, we use the formula:

\[ P(\text{frequently or occasionally}) = \frac{\text{Number of frequently involved} + \text{Number of occasionally involved}}{\text{Total number of people}} = \frac{432 + 889}{2866} \]

Calculating this gives:

\[ P(\text{frequently or occasionally}) \approx 0.4609 \]

Next, we calculate the probability that a person is female or not involved in charity work at all:

\[ P(\text{female or not involved}) = \frac{\text{Number of females} + \text{Number of not involved} - \text{Number of females not involved}}{\text{Total number of people}} = \frac{1381 + 1545 - 746}{2866} \]

Calculating this gives:

\[ P(\text{female or not involved}) \approx 0.7606 \]

Finally, we find the probability that a person is male or frequently involved in charity work:

\[ P(\text{male or frequently involved}) = \frac{\text{Number of males} + \text{Number of frequently involved} - \text{Number of males frequently involved}}{\text{Total number of people}} = \frac{1485 + 432 - 227}{2866} \]

Calculating this gives:

\[ P(\text{male or frequently involved}) \approx 0.5897 \]

Final Answer