To find [tex]\( P(Y \mid B) \)[/tex] using the information in the table, we need to understand and apply the concept of conditional probability. The conditional probability [tex]\( P(Y \mid B) \)[/tex] represents the probability of event [tex]\( Y \)[/tex] occurring given that event [tex]\( B \)[/tex] has already occurred. The formula for conditional probability is given by:
[tex]\[ P(Y \mid B) = \frac{P(Y \cap B)}{P(B)} \][/tex]
Here's a step-by-step solution:
1. Identify the required values from the table:
- We need the total number of occurrences of event [tex]\( B \)[/tex], denoted as [tex]\( P(B) \)[/tex].
- We need the number of occurrences where both [tex]\( Y \)[/tex] and [tex]\( B \)[/tex] happen simultaneously, denoted as [tex]\( P(Y \cap B) \)[/tex].
2. Extract the values from the table:
- Looking at row [tex]\( B \)[/tex]: the total number of occurrences of [tex]\( B \)[/tex] is 85.
- The number of occurrences where [tex]\( Y \)[/tex] and [tex]\( B \)[/tex] both occur is 34.
3. Calculate the probability:
[tex]\[ P(Y \mid B) = \frac{P(Y \cap B)}{P(B)} = \frac{34}{85} \][/tex]
4. Simplify the fraction:
[tex]\[ \frac{34}{85} = \frac{34 \div 17}{85 \div 17} = \frac{2}{5} = 0.4 \][/tex]
5. Round to the nearest tenth:
Although our result, [tex]\( 0.4 \)[/tex], is already at a precision of one decimal place, observe that rounding [tex]\( 0.4 \)[/tex] to the nearest tenth remains [tex]\( 0.4 \)[/tex].
Therefore, to the nearest tenth, the value of [tex]\( P(Y \mid B) \)[/tex] is:
[tex]\[ \boxed{0.4} \][/tex]
