Question 1200213
Certainly, let's solve this problem using a Venn diagram.

**1. Define Events:**

* **R1:** Event of drawing a red card from Deck X.
* **B1:** Event of drawing a black card from Deck X.
* **R2:** Event of drawing a red card from Deck Y (after the transfer).
* **B2:** Event of drawing a black card from Deck Y (after the transfer).

**2. Calculate Initial Probabilities:**

* **P(R1) = 8/15** (Probability of drawing a red card from Deck X initially)
* **P(B1) = 7/15** (Probability of drawing a black card from Deck X initially)

**3. Determine Probabilities after Transferring Cards:**

* If a red card is drawn from Deck X (R1):
    * Deck Y will have 7 red cards and 6 black cards.
    * P(R2 | R1) = 7/13 
    * P(B2 | R1) = 6/13

* If a black card is drawn from Deck X (B1):
    * Deck Y will have 8 red cards and 5 black cards.
    * P(R2 | B1) = 8/13
    * P(B2 | B1) = 5/13

**4. Calculate Probabilities Using Total Probability Theorem:**

* P(R2) = P(R1) * P(R2 | R1) + P(B1) * P(R2 | B1)
* P(R2) = (8/15) * (7/13) + (7/15) * (8/13)
* P(R2) = 56/195 + 56/195
* P(R2) = 112/195

**5. Create the Venn Diagram:**

* **Region 1 (R2 ∩ R1):** 
    * Represents the event of drawing a red card from Deck X (R1) and then a red card from Deck Y (R2).
    * Probability: P(R1) * P(R2 | R1) = (8/15) * (7/13) = 56/195

* **Region 2 (R2 ∩ B1):** 
    * Represents the event of drawing a black card from Deck X (B1) and then a red card from Deck Y (R2).
    * Probability: P(B1) * P(R2 | B1) = (7/15) * (8/13) = 56/195

* **Region 3 (B2 ∩ R1):** 
    * Represents the event of drawing a red card from Deck X (R1) and then a black card from Deck Y (B2).
    * Probability: P(R1) * P(B2 | R1) = (8/15) * (6/13) = 48/195

* **Region 4 (B2 ∩ B1):** 
    * Represents the event of drawing a black card from Deck X (B1) and then a black card from Deck Y (B2).
    * Probability: P(B1) * P(B2 | B1) = (7/15) * (5/13) = 35/195

**Venn Diagram:**

```
     R2            B2
      |              |
  -----|--------------|-----
  | 56/195 | 48/195 |
  -----|--------------|-----
  | 56/195 | 35/195 |
      |              |
     R1            B1 
```

**Therefore, the Venn diagram with the probabilities for each region is as shown above.**

Let me know if you have any other questions or would like to explore further!