If M and N are any two events, the probability that the exactly one of them occurs is

Options

A. `P(M)+P(N)-2P(McapN)`
B. `P(M)+P(N)-P(M capN)`
C. `P(M^(C))+P(N^(C))-2P(M^(C)capN^(C))`
D. `P(McapN^(C))+P(M^(C)capN)`

Answer

To find the probability that exactly one of the two events M and N occurs, we can use the following approach:

### Step 1: Understand the Events
We need to determine the probability that either event M occurs or event N occurs, but not both. This can be expressed mathematically as:
\[ P(\text{exactly one of } M \text{ or } N) = P(M \cap N^c) + P(M^c \cap N) \]
Where:
- \( M^c \) is the complement of event M (M does not occur).
- \( N^c \) is the complement of event N (N does not occur).

### Step 2: Apply the Definition of Probability
Using the definition of probability for the intersection of events, we can rewrite the above expression:
\[ P(M \cap N^c) = P(M) - P(M \cap N) \]
\[ P(M^c \cap N) = P(N) - P(M \cap N) \]

### Step 3: Combine the Probabilities
Now, substituting these into our equation gives:
\[ P(\text{exactly one of } M \text{ or } N) = (P(M) - P(M \cap N)) + (P(N) - P(M \cap N)) \]
This simplifies to:
\[ P(\text{exactly one of } M \text{ or } N) = P(M) + P(N) - 2P(M \cap N) \]

### Step 4: Final Expression
Thus, the probability that exactly one of the events M or N occurs is:
\[ P(\text{exactly one of } M \text{ or } N) = P(M) + P(N) - 2P(M \cap N) \]

### Summary
The final formula for the probability that exactly one of the two events occurs is:
\[ P(\text{exactly one of } M \text{ or } N) = P(M) + P(N) - 2P(M \cap N) \]

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