Answered

If $A$ and $B$ are mutually exclusive, then ____.


A$P(A \cap B)=0$
B$P(A)+P(B)=1$
C$P(A \cap B)=1$
D$P(A)+P(B)=0$

Verified Answer :

  • Mutually exclusive events by definition cannot occur simultaneously, meaning their intersection is the empty set: \(A \cap B = \emptyset\).
  • The probability of an impossible event is zero, so \(P(A \cap B) = P(\emptyset) = 0\).
  • Comparing this to the provided choices, the correct property is \(\boxed{P(A \cap B) = 0}\).

Explanation

  1. Understand the definition of mutually exclusive events

The term 'mutually exclusive' in probability describes two events that cannot happen at the same time. If event \(A\) occurs, event \(B\) cannot occur, and vice versa. In terms of set theory, this means that the intersection of the sets representing events \(A\) and \(B\) contains no elements. Mathematically, we express this as:
\[A \cap B = \emptyset\]
where \(\emptyset\) is the empty set.

  1. Relate the set-theoretic definition to probability

In probability theory, the probability of an event is defined as the measure of how likely that event is to occur. An impossible event is one that has no chance of occurring, and its probability is always assigned a value of \(0\). Since the intersection \(A \cap B\) represents the event that both \(A\) and \(B\) occur simultaneously, and we know that this is impossible for mutually exclusive events, it follows that:
\[P(A \cap B) = P(\emptyset) = 0\]

  1. Evaluate the options provided

Now we examine the given options:

  • \(P(A \cap B) = 0\): This matches our derived definition.
  • \(P(A) + P(B) = 1\): This is the definition of complementary events (where \(B\) is the exact opposite of \(A\)), not necessarily mutually exclusive events.
  • \(P(A \cap B) = 1\): This would imply that both events happen with certainty, which is the opposite of being mutually exclusive.
  • \(P(A) + P(B) = 0\): Since individual probabilities must be greater than or equal to \(0\), this would only be true if both events were impossible themselves, which is not a requirement for mutual exclusivity.

Thus, the only correct option is the first one.

Examples

In real life, consider the act of rolling a single six-sided die. Let event \(A\) be rolling an even number \(\{2, 4, 6\}\) and event \(B\) be rolling the number \(1\). Because it is impossible to roll a number that is both even and \(1\) at the same time, these events are mutually exclusive. Therefore, the probability of rolling a number that is both even and \(1\) is \(P(A \cap B) = 0\). This concept is fundamental in calculating the probability of the union of two events, as it simplifies the addition rule to \(P(A \cup B) = P(A) + P(B) - P(A \cap B) = P(A) + P(B)\).

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