A mail-order firm considers three possible events in filling...

A mail-order firm considers three possible events in filling an order: $A$ : The wrong item is sent. $B$ : The item is lost in transit. C: The item is damaged in transit. Assume that $A$ is independent of both $B$ and $C$ and that $B$ and $C$ are mutually exclusive. The individual event probabilities are $P(A)=0.02, P(B)=0.01$, and $P(C)=0.04$. Find the probability that at least one of these foul-ups occurs for a randomly chosen order.

A mail-order firm considers three possible events in filling an order:
$A$ : The wrong item is sent.
$B$ : The item is lost in transit.
C: The item is damaged in transit.
Assume that $A$ is independent of both $B$ and $C$ and that $B$ and $C$ are mutually exclusive. The individual event probabilities are $P(A)=0.02, P(B)=0.01$, and $P(C)=0.04$. Find the probability that at least one of these foul-ups occurs for a randomly chosen order.

Key Concepts

Union of Events

The union of events refers to the occurrence of at least one among multiple specified events. In probability theory, calculating the union involves determining the probability that one or more of the events occurs, often requiring careful handling of overlaps between events.

Inclusion-Exclusion Principle

The inclusion-exclusion principle is a method used to compute the probability of the union of multiple events by first summing the probabilities of the individual events, then subtracting the probabilities of all pairwise intersections, and adding back the probabilities of triple intersections, and so on. This technique adjusts for the over-counting that happens when events are not mutually exclusive.

Independence of Events

Independence of events occurs when the occurrence of one event does not affect the probability of another. In problems where events are independent, the probability of their intersection is the product of their individual probabilities. This property simplifies calculations and is fundamental in many probability scenarios.

Mutually Exclusive Events

Mutually exclusive events are events that cannot occur simultaneously. When two events are mutually exclusive, the probability of their intersection is zero. This concept is critical when applying the inclusion-exclusion principle since overlapping probabilities are absent, thereby simplifying the union probability calculation.

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