The question asks for the probability that at least two of the three events A, B, and C occur together. This means we are interested in the outcomes where exactly two events occur or where all three events occur.

Let's break down what "at least two events occur together" means in terms of set theory:

• Event A and Event B occur (and C may or may not): This is represented by the intersection \( A \cap B \).
• Event B and Event C occur (and A may or may not): This is represented by the intersection \( B \cap C \).
• Event C and Event A occur (and B may or may not): This is represented by the intersection \( C \cap A \).

The event "at least two events occur together" is the union of these pairwise intersections: \( (A \cap B) \cup (B \cap C) \cup (C \cap A) \). We need to find the probability of this union, \( P((A \cap B) \cup (B \cap C) \cup (C \cap A)) \).

We can use the Principle of Inclusion-Exclusion for the union of three sets. For any three sets X, Y, and Z, the probability of their union is given by:

\( P(X \cup Y \cup Z) = P(X) + P(Y) + P(Z) - P(X \cap Y) - P(Y \cap Z) - P(Z \cap X) + P(X \cap Y \cap Z) \)

In our case, we let \( X = A \cap B \), \( Y = B \cap C \), and \( Z = C \cap A \). Now we need to find the intersections of these sets:

• The intersection of X and Y: \( X \cap Y = (A \cap B) \cap (B \cap C) \). Since intersection is associative and idempotent, this simplifies to \( A \cap B \cap C \).
• The intersection of Y and Z: \( Y \cap Z = (B \cap C) \cap (C \cap A) \). This simplifies to \( A \cap B \cap C \).
• The intersection of Z and X: \( Z \cap X = (C \cap A) \cap (A \cap B) \). This simplifies to \( A \cap B \cap C \).
• The intersection of X, Y, and Z: \( X \cap Y \cap Z = (A \cap B) \cap (B \cap C) \cap (C \cap A) \). This also simplifies to \( A \cap B \cap C \).

Now, substituting these simplified intersections back into the Inclusion-Exclusion Principle formula:

\( P((A \cap B) \cup (B \cap C) \cup (C \cap A)) = P(A \cap B) + P(B \cap C) + P(C \cap A) \) \( \phantom{= P(A \cap B) + P(B \cap C) + P(C \cap A)} - P(A \cap B \cap C) - P(A \cap B \cap C) - P(A \cap B \cap C) \) \( \phantom{= P(A \cap B) + P(B \cap C) + P(C \cap A) - P(A \cap B \cap C) - P(A \cap B \cap C) - P(A \cap B \cap C)} + P(A \cap B \cap C) \)

Combining the terms involving \( P(A \cap B \cap C) \):

\( -P(A \cap B \cap C) - P(A \cap B \cap C) - P(A \cap B \cap C) + P(A \cap B \cap C) = -3P(A \cap B \cap C) + P(A \cap B \cap C) = -2P(A \cap B \cap C) \)

So, the probability that at least two events occur together is:

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

Let's compare this result with the given options:

Our derived formula matches option 3.

Revision Table: Probability of Events

Additional Information: Understanding Event Probabilities

When dealing with probabilities of multiple events, understanding how events combine is crucial. The terms 'and' and 'or' are key:

• 'And' (\( \cap \)): This refers to the intersection of events. \( A \cap B \) means both A and B must happen. The probability \( P(A \cap B) \) is involved in calculating joint probabilities. If events are independent, \( P(A \cap B) = P(A)P(B) \). If dependent, conditional probability is used.
• 'Or' (\( \cup \)): This refers to the union of events. \( A \cup B \) means A happens, or B happens, or both happen. The probability \( P(A \cup B) \) is calculated using the Inclusion-Exclusion Principle to avoid double-counting the intersection.

The event "at least two of A, B, C occur" is a complex event. It includes the cases where exactly two happen (A and B but not C, A and C but not B, B and C but not A) and the case where all three happen (A and B and C). Using the union of pairwise intersections \( (A \cap B) \cup (B \cap C) \cup (C \cap A) \) correctly captures all these scenarios. \( A \cap B \) already includes the case where C also happens, which is why using the union of these pairwise intersections and the Inclusion-Exclusion principle is the correct method.