This question involves filling in the missing probabilities on a tree diagram and calculating the joint probabilities for each outcome. Let’s break it down step by step:

---

Step 1: Understanding the tree diagram structure

1. Stage 1:

   • There are two nodes, A and B.
   • The probability of reaching A is $P(A) = 0.7$, so the probability of reaching B must be $P(B) = 1 - P(A) = 0.3$.

2. Stage 2:

   • From node A, there are two branches leading to outcomes C and D.

     • The probability of A → C is $P(C|A) = 0.4$.
     • Therefore, the probability of A → D must be $P(D|A) = 1 - P(C|A) = 0.6$.

   • From node B, there are two branches leading to outcomes C and D.

     • The probability of B → D is $P(D|B) = 0.6$.
     • Therefore, the probability of B → C must be $P(C|B) = 1 - P(D|B) = 0.4$.

---

Step 2: Fill in missing probabilities

1. $P(B) = 0.3$
2. $P(D|A) = 0.6$
3. $P(C|B) = 0.4$

---

Step 3: Compute joint probabilities for outcomes

The joint probabilities $P(A ∩ C)$, $P(A ∩ D)$, $P(B ∩ C)$, and $P(B ∩ D)$ are calculated by multiplying the probabilities along each path:

1. $P(A ∩ C) = P(A) \times P(C|A)$
   $= 0.7 \times 0.4 = 0.28$

2. $P(A ∩ D) = P(A) \times P(D|A)$
   $= 0.7 \times 0.6 = 0.42$

3. $P(B ∩ C) = P(B) \times P(C|B)$
   $= 0.3 \times 0.4 = 0.12$

4. $P(B ∩ D) = P(B) \times P(D|B)$
   $= 0.3 \times 0.6 = 0.18$

---

Final Answers:

• $P(A ∩ C) = 0.28$
• $P(A ∩ D) = 0.42$
• $P(B ∩ C) = 0.12$
• $P(B ∩ D) = 0.18$

---

Would you like a visual of the tree diagram? 😊 Let me know!

---

Related Questions:

1. How do you interpret the joint probability $P(A ∩ C)$?
2. What does it mean when $P(B ∩ D)$ is greater than $P(B ∩ C)$?
3. How do conditional probabilities affect the calculation of joint probabilities?
4. What are the rules for ensuring all probabilities on a tree diagram sum to 1?
5. How can you extend this problem to a tree diagram with more than two stages?

Tip: Always check that the sum of probabilities from each node in a tree diagram equals 1. This ensures consistency and correctness.