If A. B. C are three events, then what is the probability that atleast two of these events occur together?

Options

A. `P(A nn B) + P( B nn C)+P(C nnA)`
B. `P(A nn B) + P(B nn C) + P(C nnA) -P(A nn B nn C)`
C. `P(A nn B) + P(B nn C) + P(C nnA) -2P(A nn B nn C)`
D. `P(A nn B) + P(B nn C) + P(C nnA) -3P(A nn B nn C)`

Answer

To find the probability that at least two of the events A, B, and C occur together, we can use the principle of inclusion-exclusion. Here’s a step-by-step solution:

### Step 1: Understand the Problem
We need to find the probability that at least two of the events A, B, and C occur together. This means we need to consider the cases where:
1. Exactly two events occur.
2. All three events occur.

### Step 2: Define the Events
Let:
- \( P(A) \) = Probability of event A occurring
- \( P(B) \) = Probability of event B occurring
- \( P(C) \) = Probability of event C occurring

We also need to consider the intersections of these events:
- \( P(A \cap B) \) = Probability that both A and B occur
- \( P(A \cap C) \) = Probability that both A and C occur
- \( P(B \cap C) \) = Probability that both B and C occur
- \( P(A \cap B \cap C) \) = Probability that all three events occur

### Step 3: Use the Inclusion-Exclusion Principle
The probability that at least two of the events occur can be calculated as follows:

\[ P(\text{at least two of A, B, C}) = P(A \cap B) + P(A \cap C) + P(B \cap C) - 2P(A \cap B \cap C) \]

### Step 4: Explanation of the Formula
- We add the probabilities of the pairs of events (A and B, A and C, B and C).
- We subtract twice the probability of all three events occurring because when we add the pairs, the case where all three events occur is counted three times (once in each pair), and we need to remove it twice to account for it being added too many times.

### Step 5: Final Expression
Thus, the final expression for the probability that at least two of the events A, B, and C occur together is:

\[ P(A \cap B) + P(A \cap C) + P(B \cap C) - 2P(A \cap B \cap C) \]