Basic Concepts of Probability
Read this section about basic concepts of probability, including spaces, and events. This section discusses set operations using Venn diagrams, including complements, intersections, and unions. Finally, it introduces conditional probability and talks about independent events.
Definition
The intersection of events and , denoted , is the collection of all outcomes that are elements of both of the sets and . It corresponds to combining descriptions of the two events using the word "and".
To say that the event occurred means that on a particular trial of the experiment both and occurred. visual representation of the intersection of events and in a sample space is given in Figure 3.4 "The Intersection of Events ". The intersection corresponds to the shaded lens-shaped region that lies within both ovals.
Figure 3.4 The Intersection of Events A and B
EXAMPLE 12
In the experiment of rolling a single die, find the intersection of the events "the number rolled is even" and : "the number rolled is greater than two".
Solution:
The sample space is . Since the outcomes that are common to and are 4 and .
In words the intersection is described by "the number rolled is even and is greater than two". The only numbers between one and six that are both even and greater than two are four and six, corresponding to given above.
EXAMPLE 13
A single die is rolled.
a. Suppose the die is fair. Find the probability that the number rolled is both even and greater than two.
b. Suppose the die has been "loaded" so that , and the remaining four outcomes are equally likely with one another. Now find the probability that the number rolled is both even and greater than two.
Solution:
In both cases the sample space is and the event in question is the intersection of the previous example.
a. Since the die is fair, all outcomes are equally likely, so by counting we have .
b. The information on the probabilities of the six outcomes that we have so far is
Since and the probabilities of all six outcomes add up to 1 ,
Thus , so . In particular . Therefore
Definition
Events and are mutually exclusive if they have no elements in common.
For and to have no outcomes in common means precisely that it is impossible for both and to occur on a single trial of the random experiment. This gives the following rule.
Probability Rule for Mutually Exclusive Events
Events and are mutually exclusive if and only if
Any event and its complement are mutually exclusive, but and can be mutually exclusive without being complements.
EXAMPLE 14
In the experiment of rolling a single die, find three choices for an event so that the events and : "the number rolled is even" are mutually exclusive.
Solution:
Since and we want to have no elements in common with , any event that does not contain any even number will do. Three choices are (the complement , the odds), , and .