Probability

Step 1: Define the events.
- is the event of getting exactly two heads. The possible outcomes are , so . - is the event of getting at most two tails. The possible outcomes are all three outcomes except , so .

Step 2: Use the formula for the union of events.
The formula for the union of two events is: The only outcome in which both events occur is , so . Thus, the correct answer is . Therefore, the correct answer is 4. .

Step 1: Define events.
Let = ball drawn from Bag A, = ball drawn from Bag B. Each bag is equally likely .

Step 2: Find probability of Red from each bag.
- From Bag A: .
- From Bag B: .

Step 3: Total probability of Red.

Step 4: Apply Bayes' theorem.

Step 5: Conclusion.
Thus, the probability that the Red ball came from Bag is .