Question 1200213: Deck X has fifteen cards of which 8 are red and 7 are black. Deck Y has twelve cards of which 6 are
red and 6 are black. A card is drawn at random from Deck X and is discarded. Then all the
remaining cards are transferred to Deck Y. A card is now drawn from Deck Y and is found to be
red.
For this problem, draw a Venn diagram and fill in each of the 4 regions with their correct
symbols and probabilities using fractions. Show complete solutions.
Answer by textot(100)   (Show Source):
You can put this solution on YOUR website! 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!
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