Basic Concepts of Probability
6.6 The Addition Rule
Theaddition ruleis used to determine the probability of occurrence of one of many possible events. It is typically applied when the question has the word“or” in it. For example,“What is the probability of rolling a die and obtaining a 4ora 6?; What is the probability of drawing a clubora heart from a deck of cards?”We can represent this concept by using the termP(A or B). [Note that in mathemat- ics, this is usually written asP(A B), but we will just use the word“or”rather
6.6 The Addition Rule 171
than introduce a new symbol.] Determining the proper formula to use when answering an“or”question depends upon another concept–mutual exclusivity.
Mutually exclusive eventsoccur when one event precludes the occurrence of another event. (The term“disjoint”may also be used to represent this concept.) For example, when we roll a die once, it is impossible to obtain a 4 and a 6; one precludes the other. When we select a card from the deck, it cannot have more than one suit (we can either get a spade, heart, club, or diamond; the suits cannot co-occur within the same elemental event). If the two events are mutually exclusive, the formula for the addition rule follows.
Addition rule formula for two mutually exclusive events P(A or B) =P(A) +P(B) (Formula 6.2)
Formula 6.2 is read as“The probability of either eventAor eventBoccurring equals the probability of event A occurring plus the probability of event Boccurring.”
■QuestionAssume that we roll one die. What is the probability of coming up with a 2 or a 5?
Solution
Step 1.First, determineP(A). Let us call rolling a 2 eventA. Is each event equally likely? Assuming the die is a fair die, we can assume this. So, we will use For- mula 6.1 to answer this question.
P A = number of favorable events total number of events =1
6= 17
The probability of rolling a 2 is .17. (This value is more accurately stated as the fraction 1/6.)
Step 2.Determine the probability of eventBoccurring. Let us call rolling a 5 eventB. Since each event is equally likely, let us use Formula 6.1 once again.
P B = number of favorable events total number of events =1
6= 17
The probability of rolling a 5 is .17. (This value is more accurately stated as the fraction 1/6.)
Step 3.Since eventsAandBare mutually exclusive, we can use Formula 6.2 to determine the probability of rolling a 2 or a 5.
P A or B =P A +P B
P A or B = 1 6 + 1 6 =2 6or 33
The probability of rolling a 2 or a 5 on a single toss of the die is .33. ■
The addition rule for mutually exclusive events is generalizable to situations where we want to determine the probability of occurrence of one of several events. The general equation for the addition rule with more than two mutually exclusive events follows.
Addition rule formula for more than two mutually exclusive events P(A or B or C or…Z) =P(A) +P(B) +P(C) + +P(Z) (Formula 6.3)
where
P(Z) =the probability of occurrence of the last event
Formula 6.3 is merely an extension of the addition rule formula for two mutually exclusive events. As a consequence, the computational steps follow the same format as outlined in the preceding worked example. Below are a couple worked examples.
■QuestionWhat is the probability of randomly selecting a 3, 7, or 9 from a deck of cards?
Solution
Step 1.First determine theP(A). Let us call drawing a 3 eventA. Since each elemental event is equally likely, we can use Formula 6.1.
P A = number of favorable events total number of events = 4
52= 0769
The probability of randomly selecting a 3 from the deck is .0769. (We should feel free to use a couple more decimal places when dealing with probabilities, since the values only range from 0 to 1.)
Step 2.Determine the probability of eventBoccurring. Let us call drawing a 7 event B. Since each elemental event is equally likely, once again we can use Formula 6.1.
P B = number of favorable events total number of events = 4
52= 0769
The probability of randomly selecting a 7 from the deck is also .0769.
Step 3.Determine the probability of eventCoccurring. Let us call drawing a 9 event C. Since each elemental event is equally likely, once again we can use Formula 6.1.
P C = number of favorable events total number of events = 4
52= 0769
The probability of randomly selecting a 9 from the deck is also .0769.
Step 4.Use the addition rule to sum the separate probabilities.
P A or B or C =P A +P B +P C P A or B or C = 769 + 769 + 769 = 23
This means the probability of selecting a 3, 7, or 9, on a single draw, is .23.■ 6.6 The Addition Rule 173
Let us try one more example. This time the problem will involve 4 elemental events, and we will take some liberties with the process.
■QuestionAssume that we roll one die. What is the probability of coming up with a number less than 5?
Solution
Step 1.Since all elemental events can be considered equally likely, and since each elemental event is mutually exclusive, we can use Formula 6.3.
P 1or2or3or4 =P 1 +P 2 +P 3 +P 4 = 1 6 + 1 6 + 1 6 + 1 6 = 4 6or 67
The probability of rolling a number less than 5 on a single toss of the die is .67.■
We must watch out since not all cases ofP(A or B) can be worked that easily;
sometimes eventsAandBcan co-occur. For example, what is the probability of drawing a heart or a queen? Well, the probability of drawing a heart is 13/52 (or 1/4), and the probability of drawing a queen is 4/52 (or 1/13). But the probability of drawing a heart or a queen is 16/52; this is not the sum of 13/52 and 4/52 (which is 17/52). What is the difference? In the first example, the two events (jack and queen) cannot both occur at the same time. However, in the second example, the two events (heart and queen) can co-occur; one can draw both a queen and a heart at the same time. In card language, that card is the queen of hearts; and in mathematical language, that card represents a co- occurrence. An event being a “heart”is not mutually exclusive from an event being a “queen.” If we leave the formula as is, the queen of hearts card will be counted twice, once when we tally up all of the queens and again when we tally up all of the hearts. We need to change our formula to keep from dou- ble-counting events that qualify as being both an event Aand eventB. In so doing we will need to introduce a new concept, the probability of both events A and Bco-occurring. (This idea will be more fully explained in the following sections of the chapter.) Here is the formula for finding the likelihood of events AorBoccurring if they are not mutually exclusive.
Addition rule formula for two events
P(A or B) =P(A) +P(B)−P(A and B) (Formula 6.4)
P(A and B) can be read as “The probability of both event A and event B co-occurring.”In the above example of selecting a queen or a heart, the probability of both a queen and a heart co-occurring is 1/52. This helps us understand why the correct answer is not 17/52, but rather 16/52. Perhaps it is helpful to point out that Formula 6.2 is just a special case of the more general Formula 6.4. Formula 6.4
always works; but Formula 6.2 works whenP(AandB) = 0. In these situations the last expression in Formula 6.4 simply falls out of the formula, leaving us with Formula 6.2.
■QuestionSuppose we are about to win a game if we roll either an even number or a number greater than 4. What are our chances of winning?
Solution
Step 1.First determine theP(A). Let us call rolling an even number eventA. Let us further assume that we know that the chance of rolling an even number is .5.
Step 2.Second determine theP(B). Let us call rolling a number greater than 4 eventB. Let us further assume that we know that the chance of rolling a number greater than 4 is .3333.
Step 3.Finally determineP(A and B). This, when translated into our problem, means the chance of rolling a number that is both an even number and a number greater than 4. Let us assume we know that the chance of this occur- ring is .1667. (The number 6 is the only value on a standard die that is both an even numberanda number that is greater than 4. Since all elemental events are equally likely, the chance of rolling a 6 is 1/6 or .1667.)
P A or B =P A +P B −P A and B P A or B = 5 + 3333− 1667 = 6667or2 3
The probability of rolling a die and getting either an even number or a number greater than 4 is .6667. We have a 2/3rd chance of winning on the next roll.■
The addition rule for more than two events when the events are not mutually exclusive is much more complicated. Since this chapter is merely an introduc- tion to probability, this topic will not be covered.