MANAGEMENT SCIENCE
Chapter 2:
Lesson 2 : Mutually and Not Mutually Exclusive Events
MUTUALLY EXCLUSIVE AND NOT MUTUALLY EXCLUSIVE EVENTS:
❖ Events are said to be mutually exclusive if only one of the events can occur on any one trial.
❖ Events are mutually exclusive if one and only one of them can take place at a time.
❖ Consider again the example of the coin toss. We have two possible outcomes, heads and tails. On any single toss,
either heads or tails may turn up, but not both.
❖ Accordingly, the events heads and tails on a single toss are said to be mutually exclusive.
❖ They are called not mutually exclusive if the list of outcomes includes every possible outcome.
❖ Many common experiences involve events that have both of these properties.
❖ In tossing a coin, for example, the possible outcomes are a head or a tail. Since both of them cannot occur on any one
toss, the outcomes head and tail are mutually exclusive. Since obtaining a head and a tail represent every possible
outcome, they are not mutually exclusive.
Example 2: Rolling A Die. Rolling a die is a simple experiment that has six possible outcomes, each listed in the following table
with the corresponding probability:
OUTCOME OF ROLL PROBABILITY
1 1/6
2 1/6
3 1/6
4 1/6
5 1/6
6 1/6
Total 1
These events are both mutually exclusive ( on any roll, only one of the six events can occur) and are also not mutually exclusive
(one of them must occur and hence they total in probability to 1).
Example 3: Drawing A Card. You are asked to draw one card from a deck of 52 playing cards. Using a logical probability
assessment, it is easy to set some of the relationships, such as:
P (drawing a 7) = 4/52 = 1/3
P (drawing a heart) = 13/52 = ¼
We also see that these events (drawing a 7 and drawing a heart) are not mutually exclusive since a 7 of hearts can be drawn.
They are also not collectively exhaustive since there are other cards in the deck besides 7s and hearts.
The table below is especially useful in helping to understand the difference between mutually exclusive and not mutually
exclusive events.
DRAWS MUTUALLY EXCLUSIVE
1. Draw a spade & a club Yes
2. Draw a face card & a number card Yes
3. Draw an ace & a 3 Yes
4. Draw a club & a non-club Yes
5. Draw a 5 & a diamond No
6. Draw a red card & a diamond No
Standard Deck of Cards (52 cards)
Spades 13 Black Cards Non-Face Cards 36
Hearts 13 Red Cards
Diamonds 13 Red Cards
Clubs 13 Black Cards
According to Number:
Even 4 (2,4,6,8,10)
Odd 5 (3,5,7,9)
Face Cards:
King : 4 Queen : 4 Jack : 4 Ace : 4
ADDING MUTUALLY EXCLUSIVE EVENTS
➢ Often we are interested in whether one event or a second event will occur. This is often called the union of two events.
When these two events are mutually exclusive, the law of addition is simply as follows:
P (event A or event B) = P (event A) + P (event B)
or, more briefly,
P (A or B) = P (A) + P ( B)
For example, we know that the events of drawing a spade or drawing a club out of a deck of cards are mutually
exclusive.
Since P(spade) = 13/52 and P (club) = 13/52, the probability of drawing either a spade or a club is:
P (spade or club) = P (spade) + P ( club)
= 13/52 + 13/52
= 26/52 = ½ = 0.50 = 50%
Another example: Drawing a King or a Queen
P (King or Queen) = P (King) + P (Queen)
= 4/52 + 4/52
= 8/52 = .15 or 15%
LAW OF ADDITION FOR EVENTS THAT ARE NOT MUTUALLY EXCLUSIVE
➢ When two events are not mutually exclusive, the equation should be:
P (event A or event B) = P (event A) + P (event B) - P (event a and event B both occurring)
In shorter form:
P (A or B) = P (A) + P ( B) - P ( A and B)
When events are mutually exclusive, the area of overlap, called the intersection, is 0.
➢ Let us consider the events drawing a 5 and drawing a diamond out of a deck of cards. These events are not mutually
exclusive, to compute for the probability of either a 5 or a diamond will be drawn, the following is the solution:
P (five or diamond) = P (five) + P (diamond) - P (five and diamond)
= 4/52 + 13/52 – 1/52
= 16/52
= 4/13 or 31%
* There is 31% probability of drawing a five or a diamond from a deck of cards.
Another example: Drawing a Jack and drawing a Club
P (Jack or Club) = P(Jack) + P(Club) – P (Jack and Club)
= 4/52 + 13/52 – 1/52
= 16/52 or 31%
* There is 31% probability of drawing a Jack or a Club from a deck of cards.
EXAMPLE: These are the experience data for 50 welders in a fabrication shop.
Years of experience Number Probability
0-2 5 5/50 = .1
3-5 10 10/50 = .2
6-8 15 15/50 = .3
More than 8 20 20/50 = .4
------------- ---------------
Total 50 50/50 = 1.00
What is the probability that a welder selected at random will have 6 or more years of experience?
P (6 or more) = P(6-8) + P(more than 8)
= .3 + .4
= .7 or 70%
❏ The Addition Rule for Events That Are Not Mutually Exclusive: If two events are not mutually exclusive, it is possible for
both events to occur together. In such cases, the addition rule must be modified.
❏ Let us use the example of a deck of cards to introduce the idea. What is the probability of drawing either an ace or a
spade from a deck of cards? Obviously the events ace and spade can occur together because we could draw the ace of
spades; thus ace and spade are not mutually exclusive. The correct equation to use for the probability of one or more of
two events that are not mutually exclusive is:
P (A or B) = P (A) + P (B) - P (A and B)
In the example:
P (Ace or Spade) = P (Ace) + P (Spade) – P( Ace and Spade)
EXAMPLE: The City Council of San Fernando is composed of the following:
Person Sex Age
1 Male 31
2 Male 33
3 Female 46
4 Female 29
5 Male 41
If the members of the council decide to elect a chairperson by random draw ( say, by drawing the names from a hat),
what is the probability that the chairperson will be either a female or over 35?
❏ What is the probability that the chairperson will be either a female or over 35?
Series of possible probabilities:
P ( Male ) = 3/5 P (Female) = 2/5
P (below 35) = 3/5 P (over 35) = 2/5
P (F & over 35) = 1/5 P (M & over 35)=1/5
P (F & below 35= 1/5 P (M& below 35)=2/5
P(female or over 35) = P(female) + P(over 35) – P(female and over 35)
= 2/5 + 2/5 - 1/5
= 3/5 = .6 or 60%
SUMMARY: KEY EQUATIONS
1. A basic statement of probability
0 < P (event) < 1
2. Law of addition for mutually exclusive events
P (A or B) = P (A) + P ( B)
3. Law of addition for events that are not mutually exclusive
P (A or B) = P (A) + P ( B) - P ( A and B)
