If A, B, and C are mutually exclusive events with P(A)=0.2,...

If $A, B,$ and $C$ are mutually exclusive events with $P(A)=0.2, P(B)=0.3,$ and $P(C)=0.4,$ determine the following probabilities: (a) $P(A \cup B \cup C)$ (b) $P(A \cap B \cap C)$ (c) $P(A \cap B)$ (d) $P[(A \cup B) \cap C]$ (e) $P\left(A^{\prime} \cap B^{\prime} \cap C^{\prime}\right)$

If $A, B,$ and $C$ are mutually exclusive events with $P(A)=0.2, P(B)=0.3,$ and $P(C)=0.4,$ determine the following probabilities:
(a) $P(A \cup B \cup C)$
(b) $P(A \cap B \cap C)$
(c) $P(A \cap B)$
(d) $P[(A \cup B) \cap C]$
(e) $P\left(A^{\prime} \cap B^{\prime} \cap C^{\prime}\right)$

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Key Concepts

Complement of an Event

The complement of an event consists of all outcomes in the sample space that are not part of the event. The probability of the complement is calculated by subtracting the probability of the event from 1. This is an essential concept when working with events and their non-occurrence.

Intersection of Events

The intersection of events refers to the set of outcomes that are common to all events being considered. For mutually exclusive events, the intersection between any two events is empty, meaning its probability is zero. This concept is crucial when determining the probability of multiple events occurring together.

Mutually Exclusive Events

Mutually exclusive events are events that cannot occur at the same time. In probability, if events are mutually exclusive, the occurrence of one event excludes the possibility of the other events occurring concurrently. This implies that the intersection of any two mutually exclusive events has a probability of zero.

Addition Rule for Mutually Exclusive Events

The addition rule for mutually exclusive events states that the probability of the union of these events is equal to the sum of their individual probabilities. Since no two events occur together, there is no overlap to subtract, making the calculation straightforward.

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