Question

Let \( E, F \) and \( G \) be three events such that \[ P(E \cap F \cap G) = 0.1, P(G \mid F) = 0.3 \, \text{and} \, P(E \mid F \cap G) = P(E \mid F). \] Then \( P(G \mid E \cap F) \) equals ................ 
 

Hint:

To calculate conditional probabilities, use the formula \( P(A \mid B) = \frac{P(A \cap B)}{P(B)} \).

Answer 1

Correct Answer: 0.25 - 0.35

Step 1: Understand the given probabilities.
We are given the following information: \[ P(E \cap F \cap G) = 0.1, P(G \mid F) = 0.3, P(E \mid F \cap G) = P(E \mid F). \] We need to find \( P(G \mid E \cap F) \).

Step 2: Apply the conditional probability formula.
The conditional probability \( P(G \mid E \cap F) \) is given by the formula: \[ P(G \mid E \cap F) = \frac{P(G \cap E \cap F)}{P(E \cap F)}. \] Using the fact that \( P(E \cap F \cap G) = 0.1 \) and \( P(G \mid F) = 0.3 \), we calculate: \[ P(E \cap F) = P(G \mid F) \times P(E \cap F) = 0.3 \times 0.1 = 0.03. \]

Step 3: Substitute values.
Now, we substitute into the conditional probability formula: \[ P(G \mid E \cap F) = \frac{0.1}{0.03} = 0.3. \]

Step 4: Conclusion.
The correct answer is 0.3.