Question

find ( p(y|b) ) from the information in the table.
to the nearest tenth, what is the value of ( p(y|b) )?

	x	y	z	total
b	6	34	45	85
c	23	56	32	111
total	37	170	117	324

\\( \bigcirc \\) 0.2
\\( \bigcirc \\) 0.3
\\( \bigcirc \\) 0.4
\\( \bigcirc \\) 0.5

Explanation:

Step1: Recall Conditional Probability Formula

The formula for conditional probability is \( P(Y|B) = \frac{P(Y \cap B)}{P(B)} \). For a contingency table, \( P(Y \cap B) \) is the number of occurrences of both \( Y \) and \( B \) divided by the total number of observations, and \( P(B) \) is the number of occurrences of \( B \) divided by the total number of observations. So, \( P(Y|B) = \frac{\text{Number of } Y \text{ and } B}{\text{Number of } B} \).

Step2: Identify Values from Table

From the table, the number of observations where both \( Y \) and \( B \) occur is 34 (the cell at row \( B \) and column \( Y \)). The number of observations for \( B \) (the total of row \( B \)) is 85.

Step3: Calculate the Probability

Substitute the values into the formula: \( P(Y|B) = \frac{34}{85} \). Simplify the fraction: \( \frac{34 \div 17}{85 \div 17} = \frac{2}{5} = 0.4 \).

Answer:

0.4