Questions: For two events, M and N, P(M)=0.5, P(N M)=0.3, and P(N M ')=0.5. Find P(M N). P(M N)= (Simplify your answer. Type an integer or a fraction.)

Solution Steps

To find \( P(M \mid N) \) using Bayes' Theorem, we need to use the formula:

\[ P(M \mid N) = \frac{P(N \mid M) \cdot P(M)}{P(N)} \]

First, we need to calculate \( P(N) \) using the law of total probability:

\[ P(N) = P(N \mid M) \cdot P(M) + P(N \mid M') \cdot P(M') \]

where \( P(M') = 1 - P(M) \).

We are given the following probabilities:

• \( P(M) = 0.5 \)
• \( P(N \mid M) = 0.3 \)
• \( P(N \mid M') = 0.5 \)

Using the complement rule, we find: \[ P(M') = 1 - P(M) = 1 - 0.5 = 0.5 \]

Using the law of total probability, we calculate \( P(N) \): \[ P(N) = P(N \mid M) \cdot P(M) + P(N \mid M') \cdot P(M') \] Substituting the values: \[ P(N) = (0.3 \cdot 0.5) + (0.5 \cdot 0.5) = 0.15 + 0.25 = 0.4 \]

Now, we apply Bayes' Theorem to find \( P(M \mid N) \): \[ P(M \mid N) = \frac{P(N \mid M) \cdot P(M)}{P(N)} \] Substituting the values: \[ P(M \mid N) = \frac{0.3 \cdot 0.5}{0.4} = \frac{0.15}{0.4} = 0.375 \]

Final Answer