To address this problem systematically, we will interpret the tree diagram based on the given probabilities and the relationships between the nodes.

Let's start with the structure based on the information:

• Stage 1:

  • The branch from the starting position to node A is labeled 0.7.
  • The branch from the starting position to node B has an unknown probability.

• Stage 2:

  • From node A:
    • The branch to node C is labeled 0.4.
    • The branch to node D is an unknown probability.
  • From node B:
    • The branch to node C has an unknown probability.
    • The branch to node D is labeled 0.6.

Conditions:

We are given the following outcomes and need to determine the missing probabilities:

• $P(A \cap C) = 0.4$
• $P(A \cap D) = 0.6$
• $P(B \cap C) = 0.4$
• $P(B \cap D) = 0.6$

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Step-by-Step Analysis:

• For $P(A \cap C) = 0.4$:
  • The probability of reaching node C from node A is given as 0.4.
  • The total probability of reaching node A in stage 1 is 0.7, and the probability of reaching node C from node A is 0.4.
  • Therefore, the joint probability $P(A \cap C) = 0.7 \times 0.4 = 0.28$.

However, we are told that $P(A \cap C) = 0.4$, which implies there might be a misunderstanding, as the values provided don't match directly with this outcome. Let's reassess the branch probabilities.

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Questions for clarification:

1. What is the probability value for the branch from node A to node D?
2. What is the probability value for the branch from node B to node C?
3. Could the outcomes for the intersections $P(A \cap C)$ and $P(B \cap C)$ be erroneous?