Let's start by clarifying what [tex]\( P(A \mid D) \)[/tex] and [tex]\( P(D \mid A) \)[/tex] represent and then we’ll calculate these probabilities step-by-step using the information in the given table.
### Step-by-Step Calculation for [tex]\( P(A \mid D) \)[/tex]
1. Understanding [tex]\( P(A \mid D) \)[/tex]:
[tex]\( P(A \mid D) \)[/tex] is the conditional probability of Event A occurring given that Event D has occurred. This is calculated as:
[tex]\[ P(A \mid D) = \frac{P(A \cap D)}{P(D)} \][/tex]
2. Identifying [tex]\( P(A \cap D) \)[/tex]:
[tex]\( P(A \cap D) \)[/tex] is the joint probability of both A and D occurring, which is given by the count of occurrences of both A and D in the table. From the table, this count is 2.
3. Identifying [tex]\( P(D) \)[/tex]:
[tex]\( P(D) \)[/tex] is the probability of Event D occurring, which is given by the total number of occurrences of D divided by the total number of events. From the table, the total number of occurrences of D is 10, and the total number of events is 17:
[tex]\[ P(D) = \frac{10}{17} \][/tex]
4. Calculating [tex]\( P(A \mid D) \)[/tex]:
Using the above, we get:
[tex]\[ P(A \mid D) = \frac{2}{10} = 0.2 \][/tex]
### Step-by-Step Calculation for [tex]\( P(D \mid A) \)[/tex]
1. Understanding [tex]\( P(D \mid A) \)[/tex]:
[tex]\( P(D \mid A) \)[/tex] is the conditional probability of Event D occurring given that Event A has occurred. This is calculated as:
[tex]\[ P(D \mid A) = \frac{P(D \cap A)}{P(A)} \][/tex]
2. Identifying [tex]\( P(D \cap A) \)[/tex]:
Note that [tex]\( P(D \cap A) = P(A \cap D) \)[/tex], which we identified earlier as 2.
3. Identifying [tex]\( P(A) \)[/tex]:
[tex]\( P(A) \)[/tex] is the probability of Event A occurring, which is given by the total number of occurrences of A divided by the total number of events. From the table, the total number of occurrences of A is 8, and the total number of events is 17:
[tex]\[ P(A) = \frac{8}{17} \][/tex]
4. Calculating [tex]\( P(D \mid A) \)[/tex]:
Using the above, we get:
[tex]\[ P(D \mid A) = \frac{2}{8} = 0.25 \][/tex]
### Conclusion
We have the following probabilities:
[tex]\[ P(A \mid D) = 0.2 \][/tex]
[tex]\[ P(D \mid A) = 0.25 \][/tex]
The calculated values for [tex]\( P(A \mid D) = 0.2 \)[/tex] and [tex]\( P(D \mid A) = 0.25 \)[/tex] show that these probabilities are not equal.
### Reason for the Difference
The reason [tex]\( P(A \mid D) \)[/tex] and [tex]\( P(D \mid A) \)[/tex] are not equal is due to the different denominators used in their calculations:
- [tex]\( P(A \mid D) \)[/tex] depends on the overall occurrences of D whereas
- [tex]\( P(D \mid A) \)[/tex] depends on the overall occurrences of A.
The total occurrences of D and A are not the same, which results in different probabilities. This exemplifies that conditional probabilities are not symmetric in general.
