At Westonci.ca, we provide clear, reliable answers to all your questions. Join our vibrant community and get the solutions you need. Get the answers you need quickly and accurately from a dedicated community of experts on our Q&A platform. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.
Sagot :
Let's delve into the concepts of conditional probability and analyze why [tex]\( P(A \mid D) \)[/tex] and [tex]\( P(D \mid A) \)[/tex] are not equal using the provided table:
[tex]\[ \begin{tabular}{|c|c|c|c|} \cline { 2 - 4 } \multicolumn{1}{c|}{} & C & D & Total \\ \hline A & 6 & 2 & 8 \\ \hline B & 1 & 8 & 9 \\ \hline Total & 7 & 10 & 17 \\ \hline \end{tabular} \][/tex]
### Step-by-Step Solution:
1. Understanding the Table:
- Rows represent events A and B.
- Columns represent events C and D.
- Cell Values represent the joint occurrences of the events.
2. Calculating [tex]\( P(A \mid D) \)[/tex]:
- [tex]\( P(A \mid D) \)[/tex] is the probability of event A occurring given that event D has occurred.
- Using the conditional probability formula: [tex]\( P(A \mid D) = \frac{P(A \cap D)}{P(D)} \)[/tex].
From the table:
- [tex]\( P(A \cap D) \)[/tex] is the number of instances where both A and D occur, which is the value at the intersection of row A and column D (2). Thus, [tex]\( P(A \cap D) = 2 \)[/tex].
- [tex]\( P(D) \)[/tex] is the total number of instances where D occurs, which is the sum of the values in column D (10). Thus, [tex]\( P(D) = 10 \)[/tex].
Now, we can calculate:
[tex]\[ P(A \mid D) = \frac{2}{10} = 0.2 \][/tex]
3. Calculating [tex]\( P(D \mid A) \)[/tex]:
- [tex]\( P(D \mid A) \)[/tex] is the probability of event D occurring given that event A has occurred.
- Using the conditional probability formula: [tex]\( P(D \mid A) = \frac{P(D \cap A)}{P(A)} \)[/tex].
From the table:
- [tex]\( P(D \cap A) \)[/tex] is the number of instances where both D and A occur, which is again the value at the intersection of row A and column D (2). Thus, [tex]\( P(D \cap A) = 2 \)[/tex].
- [tex]\( P(A) \)[/tex] is the total number of instances where A occurs, which is the sum of the values in row A (8). Thus, [tex]\( P(A) = 8 \)[/tex].
Now, we can calculate:
[tex]\[ P(D \mid A) = \frac{2}{8} = 0.25 \][/tex]
4. Comparison and Explanation:
- When we compare the two probabilities [tex]\( P(A \mid D) \)[/tex] and [tex]\( P(D \mid A) \)[/tex]:
[tex]\[ P(A \mid D) = 0.2 \quad \text{and} \quad P(D \mid A) = 0.25 \][/tex]
- These probabilities are not equal because they refer to different conditional scenarios. Specifically:
- [tex]\( P(A \mid D) \)[/tex] considers the fraction of A within the context of D's occurrences.
- [tex]\( P(D \mid A) \)[/tex] considers the fraction of D within the context of A's occurrences.
The denominators in these calculations reflect different contexts: total occurrences of D for [tex]\( P(A \mid D) \)[/tex] and total occurrences of A for [tex]\( P(D \mid A) \)[/tex]. Thus, the probabilities are computed with different bases, resulting in different values.
### Conclusion:
The probabilities [tex]\( P(A \mid D) \)[/tex] and [tex]\( P(D \mid A) \)[/tex] are not equal because they refer to different conditional probabilities and are calculated with different denominators. This difference in context and base of calculation leads to different results.
[tex]\[ \begin{tabular}{|c|c|c|c|} \cline { 2 - 4 } \multicolumn{1}{c|}{} & C & D & Total \\ \hline A & 6 & 2 & 8 \\ \hline B & 1 & 8 & 9 \\ \hline Total & 7 & 10 & 17 \\ \hline \end{tabular} \][/tex]
### Step-by-Step Solution:
1. Understanding the Table:
- Rows represent events A and B.
- Columns represent events C and D.
- Cell Values represent the joint occurrences of the events.
2. Calculating [tex]\( P(A \mid D) \)[/tex]:
- [tex]\( P(A \mid D) \)[/tex] is the probability of event A occurring given that event D has occurred.
- Using the conditional probability formula: [tex]\( P(A \mid D) = \frac{P(A \cap D)}{P(D)} \)[/tex].
From the table:
- [tex]\( P(A \cap D) \)[/tex] is the number of instances where both A and D occur, which is the value at the intersection of row A and column D (2). Thus, [tex]\( P(A \cap D) = 2 \)[/tex].
- [tex]\( P(D) \)[/tex] is the total number of instances where D occurs, which is the sum of the values in column D (10). Thus, [tex]\( P(D) = 10 \)[/tex].
Now, we can calculate:
[tex]\[ P(A \mid D) = \frac{2}{10} = 0.2 \][/tex]
3. Calculating [tex]\( P(D \mid A) \)[/tex]:
- [tex]\( P(D \mid A) \)[/tex] is the probability of event D occurring given that event A has occurred.
- Using the conditional probability formula: [tex]\( P(D \mid A) = \frac{P(D \cap A)}{P(A)} \)[/tex].
From the table:
- [tex]\( P(D \cap A) \)[/tex] is the number of instances where both D and A occur, which is again the value at the intersection of row A and column D (2). Thus, [tex]\( P(D \cap A) = 2 \)[/tex].
- [tex]\( P(A) \)[/tex] is the total number of instances where A occurs, which is the sum of the values in row A (8). Thus, [tex]\( P(A) = 8 \)[/tex].
Now, we can calculate:
[tex]\[ P(D \mid A) = \frac{2}{8} = 0.25 \][/tex]
4. Comparison and Explanation:
- When we compare the two probabilities [tex]\( P(A \mid D) \)[/tex] and [tex]\( P(D \mid A) \)[/tex]:
[tex]\[ P(A \mid D) = 0.2 \quad \text{and} \quad P(D \mid A) = 0.25 \][/tex]
- These probabilities are not equal because they refer to different conditional scenarios. Specifically:
- [tex]\( P(A \mid D) \)[/tex] considers the fraction of A within the context of D's occurrences.
- [tex]\( P(D \mid A) \)[/tex] considers the fraction of D within the context of A's occurrences.
The denominators in these calculations reflect different contexts: total occurrences of D for [tex]\( P(A \mid D) \)[/tex] and total occurrences of A for [tex]\( P(D \mid A) \)[/tex]. Thus, the probabilities are computed with different bases, resulting in different values.
### Conclusion:
The probabilities [tex]\( P(A \mid D) \)[/tex] and [tex]\( P(D \mid A) \)[/tex] are not equal because they refer to different conditional probabilities and are calculated with different denominators. This difference in context and base of calculation leads to different results.
Thank you for visiting our platform. We hope you found the answers you were looking for. Come back anytime you need more information. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Stay curious and keep coming back to Westonci.ca for answers to all your burning questions.
