Find $P(C \mid Y)$ from the information in the table.
| | $X$ | $Y$ | $Z$ | Total |
| :---- | :-- | :-- | :-- | :---- |
| $A$ | 32 | 10 | 28 | 70 |
| $B$ | 6 | 5 | 25 | 36 |
| $C$ | 18 | 15 | 7 | 40 |
| Total | 58 | 30 | 60 | 148 |
To the nearest tenth, what is the value of $P(C \mid Y)$?
Answer (1)
Use the conditional probability formula: P ( C ∣ Y ) = P ( Y ) P ( C bi g c a p Y ) .
Find P ( C bi g c a p Y ) from the table: P ( C bi g c a p Y ) = 148 15 .
Find P ( Y ) from the table: P ( Y ) = 148 30 .
Calculate P ( C ∣ Y ) : P ( C ∣ Y ) = 148 30 148 15 = 0.5 .
Explanation
Understand the problem and provided data We are given a table with data about categories A, B, and C and variables X, Y, and Z. The table also includes the totals for each category and variable. We are asked to find the conditional probability P ( C ∣ Y ) , which reads as 'the probability of C given Y'. This means, given that event Y has occurred, what is the probability of event C also occurring?
Recall the formula for conditional probability To calculate P ( C ∣ Y ) , we use the formula for conditional probability: P ( C ∣ Y ) = P ( Y ) P ( C bi g c a p Y ) where P ( C bi g c a p Y ) is the probability of both C and Y occurring, and P ( Y ) is the probability of Y occurring.
Find P ( C bi g c a p Y ) from the table From the table, we can find the number of observations where both C and Y occur. This is the intersection of C and Y, which is given as 15. The total number of observations is 148. Therefore, P ( C bi g c a p Y ) = 148 15
Find P ( Y ) from the table Next, we need to find P ( Y ) . From the table, the total number of observations for variable Y is 30. The total number of observations is 148. Therefore, P ( Y ) = 148 30
Calculate P ( C ∣ Y ) Now we can substitute these values into the conditional probability formula: P ( C ∣ Y ) = 148 30 148 15 = 30 15 = 2 1 = 0.5
State the final answer The value of P ( C ∣ Y ) is 0.5. To the nearest tenth, the value remains 0.5.
Examples
Conditional probability is used in various real-life scenarios. For example, in medical diagnosis, it helps determine the probability of a disease given certain symptoms. In marketing, it helps predict the likelihood of a customer buying a product given their demographics. In finance, it's used to assess the risk of an investment given market conditions. Understanding conditional probability allows for better decision-making in situations where outcomes depend on prior events or conditions.
