- Calculate probabilities using the additive rule.
- Calculate probabilities using the multiplicative rule.
- Identify events that are not mutually exclusive and how to represent them in a Venn diagram.
- Understand the condition of independence.
When the probabilities of certain events are known, we can use those probabilities to calculate the probabilities of their respective unions and intersections. We use two rules: the additive and the multiplicative rules to find those probabilities. The examples that follow will illustrate how we can do so.
Suppose we have a loaded (unfair) die. We toss it several times and record the outcomes. If we define the following events:
Let us suppose that we have come up with and . We want to find .
It is probably best to draw the Venn diagram to illustrate the situation. As you can see, the probability of the events and occurring is the union of the individual probabilities in each event.
If we add the probabilities of and , we get
However, , thus
What we have demonstrated is that the probability of the union of two events, and , can be obtained by adding the individual probabilities of and and subtracting . The Venn diagram above illustrates this union. Formula (1) above is called the Additive Rule of Probability.
Additive Rule of Probability
The union of two events, and , can be obtained by adding the individual probabilities of and and subtracting . The Venn diagram above illustrates this union.
We can rephrase the definition as follows: The probability that either event or event occurs is equal to the probability that event occurs plus the probability that event occurs minus the probability that both occur.
Consider the experiment of randomly selecting a card from a deck of playing cards. What is the probability that the card selected is either a spade or a face card?
Our event is
The event consists of ; namely, spade cards and face cards that are not spade. Be careful, if we say that we have face cards, we would be over counting the face-spade cards!
To find we use the additive rules of probability. First, let
Note that . Remember, event consists of and event consists of face cards. Event consists of the face-spade cards: The king, jack and, queen of spades cards. Using the additive rule of probability formula,
I hope that you have learned, through this example, the reason why we subtract . It is because we do not want to count the face-spade cards twice.
If you know that of the people arrested in the mid 1990’s were males, are under the age of , and were males under . What is the probability, that a person selected at random from all those arrested, is either male or under ?
From the percents given,
The probability of a person selected is male or under 18 :
This means that of the people arrested in the mid 1990’s are either males or under .
It happens sometimes that contains no simple events, i.e., , the empty set. In this case, we say that the events and are mutually exclusive.
If contains no simple events, then and are mutually exclusive.
The figure below is the Venn diagram of mutually exclusive events, for example set might represent all the outcomes of drawing a card, and set might represent all the outcomes of tossing three coins.
This figure shows that the events and have no simple events in common, that is, events and can not occur simultaneously, and therefore,
If the events and are mutually exclusive, then the probability of the union of and is the sum of the probabilities of and , that is
Notice that since the two events are mutually exclusive, there is no over-counting.
If two coins are tossed, what is the probability of observing at least one head?
Recall from previous section that the conditional probability rule is used to compute the probability of an event, given that another event had already occurred. The formula is
Solving for , we get
This result is the Multiplicative Rule of Probability.
Multiplicative Rule of Probability
If and are two events, then
This says that the probability that both and occur equals to the probability that occurs times the conditional probability that occurs, given that occurs.
Keep in mind that the conditional probability and the multiplicative rule of probability are simply variations of the same thing.
In a certain city in the US some time ago, of all employed female workers were white-collar workers. If of all employed at the city government were female, what is the probability that a randomly selected employed worker would have been a female white-collar worker?
We first define the following events
We are seeking to find the probability of randomly selecting a female worker who is also a white-collar worker. This can be expressed as .
According to the given data, we have
Now using the multiplicative rule of probability we get,
Thus of all employed workers were white-collar female workers.
A college class has students of which are males and are females. Suppose the teacher selects two students at random from the class. Assume that the first student who is selected is not returned to the class population. What is the probability that the first student selected is a female and the second is male?
Here we may define two events
In this problem, we have a conditional probability situation. We want to determine the probability that the first student is female and the second student selected is male.
To do so we apply the multiplicative rule,
Before we use this formula, we need to calculate the probability of randomly selecting a female student from the population.
Now given that the first student is selected and not returned back to the population, the remaining number of students now is , of which female students and male students. Thus the conditional probability that a male student is selected, given that the first student selected is a female,
Substituting these values into our equation, we get
We conclude that there is a probability of that the first student selected is a female and the second one is a male.
Suppose a coin was tossed twice and the observed face was recorded on each toss. The following events are defined
Does knowing that event has occurred affect the probability of the occurrence of ?
You would probably say no. Let’s see if this is so. The sample space of this experiment is
Each of these simple events has a probability of . Looking back at the problem, we have events and .
Since the first toss is a head, we have
And since the second toss is also a head,
Now, what is the conditional probability? Here it is,
What does this tell us? It tells us that and also. Which means knowing that the first toss resulted in a head does not affect the probability of the second toss. In other words,
When this occurs, we say that events and are independent.
Condition of Independence
If event is independent of event , then the occurrence of does not affect the probability of the occurrence of event . So we write,
The table below gives the number of physicists (in thousands) in the US cross classified by specialties and base of practice . (Remark: The numbers are absolutely hypothetical and do not reflect the actual numbers in the three bases.)
Suppose a physicist is selected at random. Is the event that the physicist selected is based in academia independent of the event that the physicist selected is a nuclear physicist?
In other words, is the event independent of ?
Figure: A table showing the number of physicists in each specialty (thousands). This data is hypothetical.
The problem may appear a little difficult at first but it is actually much simpler, especially, if we make use of the condition of independence. All we need to do is to calculate and . If those two probabilities are equal, then the two events and are indeed independent. Otherwise, they are dependent. From the table we find,
Thus, and so the event is dependent on the event . This lack of independence results from the fact that the percentage of nuclear physicists who are working in the industry is not the same as the percentage of all physicists who are in the industry .
- The Additive Rule of Probability states that the union of two events can be found by adding the probabilities of each event and subtracting the intersection of the two events, or .
- If contains no simple events, then and are mutually exclusive. Mathematically, this means .
- The Multiplicative Rule of Probability states .
- If event is independent of event , then the occurrence of does not affect the probability of the occurrence of event . Mathematically, .
- Two fair dice are tossed and the following events are identified:
- Are events and independent? Why?
- Are events and mutually exclusive? Why?
- The probability that a certain brand of television fails when first used is If it does not fail immediately, the probability that it will work properly for is What is the probability that a new television of the same brand will last ?