The problem on the image is asking for the conditional probability $P(M|B)$, which means the probability of event $M$ occurring given that event $B$ has occurred.

Step-by-Step Solution:

1. Identify given data from the Venn diagram:

   • The number in $B$ only: $300$
   • The number in $M$ only: $200$
   • The number in both $B$ and $M$ (the intersection of $B$ and $M$): $50$
   • The total number of elements in the universal set (the entire box): $1000$

2. Calculate the probability of $B$ (event $B$ occurring): $P(B) = \frac{\text{Number in } B}{\text{Total number}} = \frac{300 + 50}{1000} = \frac{350}{1000} = 0.35$

3. Calculate the probability of both $B$ and $M$ (intersection): $P(B \cap M) = \frac{\text{Number in both } B \text{ and } M}{\text{Total number}} = \frac{50}{1000} = 0.05$

4. Apply the formula for conditional probability: $P(M|B) = \frac{P(B \cap M)}{P(B)} = \frac{0.05}{0.35} \approx 0.14$

Answer:

The probability of $M$ given $B$ ( $P(M|B)$ ) is approximately 0.14.

Would you like further details on this solution?

Here are 5 related questions you might find interesting:

1. What is the probability of $B$ given $M$ ( $P(B|M)$ ) using the same data?
2. How would you calculate the probability of $M$ or $B$ occurring?
3. What is the probability of neither $M$ nor $B$ occurring?
4. How does the interpretation of conditional probability change when there is no overlap between $M$ and $B$?
5. Can we verify the total probability adds up to 1 using these values?

Tip: Conditional probability often depends on the overlap between events, which is visually represented in Venn diagrams for clarity.