Question

the table below shows the results of a survey that asked 1057 adults from a certain country if they favored or opposed a tax to fund education. a person is selected at random. complete parts (a) through (c).

(a) find the probability that the person opposed the tax or is female.
\\(p(\text{opposed the tax or is female}) = \square\\)
(round to the nearest thousandth as needed.)

(b) find the probability that the person supports the tax or is male.
\\(p(\text{supports the tax or is male}) = \square\\)
(round to the nearest thousandth as needed.)

(c) find the probability that the person is not unsure or is female.
\\(p(\text{is not unsure or is female}) = \square\\)
(round to the nearest thousandth as needed.)

Explanation:

Identify the given values from the contingency table

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

• Total population \(N = 1057\)
• Total Males = \(490\), Total Females = \(567\)
• Total Support = \(408\), Total Oppose = \(613\), Total Unsure = \(36\)
• Joint frequencies:
• Males and Support = \(166\)
• Males and Oppose = \(316\)
• Males and Unsure = \(8\)
• Females and Support = \(242\)
• Females and Oppose = \(297\)
• Females and Unsure = \(28\)

Calculate probability for part (a)

Using the Addition Rule for Probability knowledge point:

$$ P(\text{Oppose} \cup \text{Female}) = P(\text{Oppose}) + P(\text{Female}) - P(\text{Oppose} \cap \text{Female}) $$

$$ P(\text{Oppose} \cup \text{Female}) = \frac{613}{1057} + \frac{567}{1057} - \frac{297}{1057} = \frac{883}{1057} \approx 0.83538 $$

Rounding to the nearest thousandth gives \(0.835\).

Calculate probability for part (b)

Using the Addition Rule for Probability knowledge point:

$$ P(\text{Support} \cup \text{Male}) = P(\text{Support}) + P(\text{Male}) - P(\text{Support} \cap \text{Male}) $$

$$ P(\text{Support} \cup \text{Male}) = \frac{408}{1057} + \frac{490}{1057} - \frac{166}{1057} = \frac{732}{1057} \approx 0.69253 $$

Rounding to the nearest thousandth gives \(0.693\).

Calculate probability for part (c)

Using the Addition Rule for Probability knowledge point:
Let \(U'\) represent "not unsure" (Support or Oppose).

• Total not unsure = \(1057 - 36 = 1021\)
• Females who are not unsure = \(242 \text{ (Support)} + 297 \text{ (Oppose)} = 539\)

$$ P(U' \cup \text{Female}) = P(U') + P(\text{Female}) - P(U' \cap \text{Female}) $$

$$ P(U' \cup \text{Female}) = \frac{1021}{1057} + \frac{567}{1057} - \frac{539}{1057} = \frac{1049}{1057} \approx 0.99243 $$

Rounding to the nearest thousandth gives \(0.992\).

Answer:

Question 1

(a) Find the probability that the person opposed the tax or is female.

\(P(\text{opposed the tax or is female}) =\) <blank>0.835</blank>

Question 2

(b) Find the probability that the person supports the tax or is male.

\(P(\text{supports the tax or is male}) =\) <blank>0.693</blank>

Question 3

(c) Find the probability that the person is not unsure or is female.

\(P(\text{is not unsure or is female}) =\) <blank>0.992</blank>