Example #1c: Two Events: Multiplication (‘And’) Law

Part c) demonstrates how to calculate the probability of the intersection of two events. If A is an event and B is another event, then P(A and B) is the probability of both A and B occurring. ‘And’ is commutative in the sense that: P(A and B) = P(B and A)

To find the probability of the intersection of two events, divide the number of outcomes that occur in both events by the number of possible outcomes. (This approach is only correct if the outcomes are equally likely.) For example, if the event is selecting a Red and an Odd marble, then:

More generally, if A and B are two events, the probability of their joint occurrence, i.e., P(A and B), is:

The following two situations simplify the multiplication law:

1. Two events are said to be mutually exclusive or disjoint if they have no outcomes in common. More specifically, if A and B are mutually exclusive events, then P(A and B) = 0. For example, P(Red and Even) = 0 since there are no marbles that are Red and have an Even number.

2. Two events are said to be independent if the occurrence of one has no effect on the probability of the occurrence of the other. More specifically, if A and B are independent events, then P(A and B) = P(A)P(B). For example, P(Blue and Even) = P(Blue)P(Even) since there are an equal number of even and odd blue marbles. A more complete discussion of independence is deferred until we examine conditional probability.

Now use the applet to compute the probabilities of other joint events and observe which marbles are in the intersections. Show that Red and Even are mutually exclusive. Show that Blue and Even are independent events.

Part a): Basic Probability

Part b): Conditioning

Part d): Or Law

Example #1