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.
LEARNING OBJECTIVES
- To learn how some events are naturally expressible in terms of other events.
- To learn how to use special formulas for the probability of an event that is expressed in terms of one or more other events.
Some events can be naturally expressed in terms of other, sometimes simpler, events.
Complements
Definition
The complement of an event in a sample space
, denoted
, is the collection of all outcomes in
that are not elements of the set
. It corresponds to negating any description in words of the event
.
EXAMPLE 10
Two events connected with the experiment of rolling a single die are : "the number rolled is even" and
: "the number rolled is greater than two". Find the complement of each.
Solution:
In the sample space the corresponding sets of outcomes are
and
. The complements are
and
.
In words the complements are described by "the number rolled is not even" and "the number rolled is not greater than two". Of course easier descriptions would be "the number rolled is odd" and "the number rolled is less than three".
If there is a 60% chance of rain tomorrow, what is the probability of fair weather? The obvious answer, 40%, is an instance of the following general rule.
Probability Rule for Complements
This formula is particularly useful when finding the probability of an event directly is difficult.
EXAMPLE 11
Find the probability that at least one heads will appear in five tosses of a fair coin.
Solution:
Identify outcomes by lists of five and
, such as
and
. Although it is tedious to list them all, it is not difficult to count them. Think of using a tree diagram to do so. There are two choices for the first toss. For each of these there are two choices for the second toss, hence
outcomes for two tosses. For each of these four outcomes, there are two possibilities for the third toss, hence
outcomes for three tosses. Similarly, there are
outcomes for four tosses and finally
outcomes for five tosses.
Let denote the event "at least one heads". There are many ways to obtain at least one heads, but only one way to fail to do so: all tails. Thus although it is difficult to list all the outcomes that form 0 , it is easy to write
. Since there are 32 equally likely outcomes, each has probability
, so
, hence
or about a
chance.