Question

students in a political science course were asked to describe their politics as \liberal\, \moderate\, or \conservative\. here are the results:

	liberal	moderate	conservative	total
male	39	53	19	111
total	66	87	34	187

given that the student is female what is the probability that she considers herself to be a \liberal\? round your answer to three decimal places if necessary.
a. 0.351 c. 0.409
b. 0.353 d. 0.355

Explanation:

Step1: Recall Conditional Probability Formula

The formula for conditional probability is \( P(A|B) = \frac{P(A \cap B)}{P(B)} \). In this case, event \( A \) is "student is Liberal" and event \( B \) is "student is female". So we need to find the number of female Liberal students and the total number of female students.

Step2: Identify the Numbers from the Table

From the table, the number of female Liberal students (\( n(A \cap B) \)) is 27, and the total number of female students (\( n(B) \)) is 76.

Step3: Calculate the Probability

Using the formula for probability (since we're dealing with counts, \( P(A|B) = \frac{n(A \cap B)}{n(B)} \)), we substitute the values: \( \frac{27}{76} \approx 0.355 \)? Wait, no, wait: \( 27\div76 \approx 0.355263\)? Wait, no, let's calculate again. Wait, 27 divided by 76: 760.35 = 26.6, 27 - 26.6 = 0.4, 0.4/76 ≈ 0.00526, so total ≈ 0.35 + 0.00526 ≈ 0.35526, which rounds to 0.355? Wait, but wait, let's do the division properly: 27 ÷ 76. Let's compute 760.355 = 760.3 + 760.05 + 760.005 = 22.8 + 3.8 + 0.38 = 26.98. 27 - 26.98 = 0.02. 0.02/76 ≈ 0.00026. So 0.355 + 0.00026 ≈ 0.35526, which is approximately 0.355. Wait, but let's check the options. Option d is 0.355. Wait, but wait, maybe I made a mistake. Wait, 27 divided by 76: 27 ÷ 76 ≈ 0.35526, which rounds to 0.355 (three decimal places). Wait, but let's check the calculation again. 760.355 = 26.98, 27 - 26.98 = 0.02, so 0.355 + 0.02/76 ≈ 0.355 + 0.00026 ≈ 0.35526, so when rounded to three decimal places, it's 0.355. Wait, but let's confirm: 27 ÷ 76 = 0.3552631579..., so to three decimal places, look at the fourth: 2, so we round the third: 5. So 0.355.

Answer:

d. 0.355