To solve this, we need to carefully analyze the given information:

• $P(A | B) = 0.1$ means the probability of $A$ occurring given $B$ occurred is 0.1.

We can explore the options step by step:

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Option a: $P(A^c | B) = 0.9$

• $P(A^c | B)$ refers to the probability that $A$ does not occur given $B$ occurred.
• From the complement rule: $P(A | B) + P(A^c | B) = 1$.
  • Here, $P(A | B) = 0.1$, so $P(A^c | B) = 1 - 0.1 = 0.9$.
• This is correct.

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Option b: $P(B | A) = 0.9$

• There's no direct relationship in the given information to calculate $P(B | A)$ from $P(A | B)$.
• This is not supported by the information given.

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Option c: $P(A^c | B^c) = 0.9$

• $P(A^c | B^c)$ refers to the probability that $A$ does not occur given $B$ does not occur.
• This requires knowledge of $P(B^c)$ or $P(A \cap B^c)$, which is not provided.
• This cannot be determined with the given information.

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Option d: $P(A | B^c) = 0.9$

• $P(A | B^c)$ refers to the probability that $A$ occurs given $B$ does not occur.
• This also requires additional data, such as $P(B^c)$ or $P(A \cap B^c)$, which is not given.
• This cannot be determined with the given information.

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Final Answer:

The correct option is a: $P(A^c | B) = 0.9$.

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Would you like a detailed breakdown of how conditional probabilities and complement rules work? Here are some related questions to explore:

1. How is $P(A^c | B)$ derived from $P(A | B)$?
2. Can we calculate $P(A | B^c)$ if we know $P(A | B)$ and $P(B)$?
3. What is the relationship between $P(A | B)$ and $P(B | A)$?
4. How does the complement rule apply in conditional probabilities?
5. What additional information is needed to determine $P(A^c | B^c)$?

Tip: Always check for relationships like complements and independence when analyzing probabilities.