### Video Transcript

In this video, we will learn how to
identify mutually exclusive events and nonβmutually exclusive events and find their
probabilities. Before we discuss mutually
exclusive events, letβs recap compound events and the addition rule of
probability.

We recall that the intersection of
events π΄ and π΅ is the collection of all outcomes that are elements of both of the
sets π΄ and π΅. And this is the equivalent to both
events occurring. The union of events π΄ and π΅ is
the collection of all outcomes that are elements of one or the other of the sets π΄
and π΅ or of both of them. This is the equivalent to either of
the events occurring.

We also recall that if an event π΄
in a sample space π cannot occur, then its probability is zero. This means that there are no
elements in π΄. And we call a set with no elements
the empty set. Finally, we recall that the
addition rule for probability states that the probability of π΄ union π΅ is equal to
the probability of π΄ plus the probability of π΅ minus the probability of π΄
intersection π΅.

Letβs now consider what happens
when the probability of this intersection equals zero. If the probability of π΄
intersection π΅ is equal to zero, then the addition rule for probability would
simplify to the probability of π΄ union π΅ is equal to the probability of π΄ plus
the probability of π΅. This leads us to an informal
definition of mutually exclusive events. Two events π΄ and π΅ where the
probability of π΄ intersection π΅ is equal to zero are mutually exclusive
events. This is because both events cannot
occur at the same time.

More formally, this can be written
as follows. π΄ and π΅ are mutually exclusive
events if π΄ intersection π΅ is equal to the empty set. This is equivalent to saying the
events cannot occur at the same time since the probability of π΄ intersection π΅ is
equal to the probability of the empty set, which we know is equal to zero.

We say that a list of events π΄ sub
one, π΄ sub two, and so on up to π΄ sub π is mutually exclusive if they are
pairwise mutually exclusive. So the intersection of π΄ sub π
and π΄ sub π is equal to the empty set for any π and π that exist in the set of
numbers one, two, and so on up to π. Summarizing, if π΄ and π΅ are
mutually exclusive, then the probability of π΄ union π΅ is equal to the probability
of π΄ plus the probability of π΅. This can be represented on a Venn
diagram as shown, where the two circles representing events π΄ and π΅ do not
intersect.

We will now consider some specific
examples. And in our first one, we will
determine whether the given pairs of events are mutually exclusive.

Amelia has a deck of 52 cards. She randomly selects one card and
considers the following events: event π΄, picking a card that is a heart; event π΅,
picking a card that is black; and event πΆ, picking a card that is not a spade. Are events π΄ and π΅ mutually
exclusive? Are events π΄ and πΆ mutually
exclusive? Are events π΅ and πΆ mutually
exclusive?

In all three parts of this
question, we need to determine whether two events are mutually exclusive. We recall that two events π₯ and π¦
are mutually exclusive if they cannot both occur at the same time, in other words,
the probability of π₯ intersection π¦ is equal to zero.

In this question, we are told that
Amelia has a regular deck of 52 cards. We know that these are split into
four suits, diamonds, hearts, clubs, and spades, where the first two suits are red
cards and the second two are black. Each of the suits has 13 cards: an
ace, the numbers two through 10, a jack, a queen, and a king.

There are three events that Amelia
needs to consider: firstly, event π΄, picking a card that is a heart. This would involve picking any of
the 13 cards in the second row. Event π΅ involves picking a card
that is black. This involves picking a card that
is either a club or a spade, any of the 26 cards in the bottom two rows. Finally, we have event πΆ, which is
picking a card that is not a spade. This could be either a diamond,
heart, or club, any of the 39 cards in the top three rows.

To determine whether events π΄ and
π΅ are mutually exclusive, we need to find whether there is an outcome that occurs
in both events. Is it possible to pick a card that
is a heart and pick a card that is black? We know that all hearts are red, so
there are no cards which are both black and a heart. And we can therefore conclude that
when Amelia is selecting one card, both events cannot happen. And the events are therefore
mutually exclusive.

Next, we need to consider whether
events π΄ and πΆ are mutually exclusive. This time, we have the events of
picking a card that is a heart and picking a card that is not a spade. The key point here is that all
hearts are not spades. This means that choosing any heart
will satisfy both events. And since both events can occur at
the same time, we can conclude that they are not mutually exclusive. There is an overlap between picking
a card that is a heart and picking a card that is not a spade.

Finally, we need to consider
whether events π΅ and πΆ are mutually exclusive. This time, we have the events of
picking a card that is black and picking a card that is not a spade. This time, the key fact is that all
clubs are black. And they are also not a spade. This means that choosing any of the
clubs satisfies both events. And we can therefore conclude that
events π΅ and πΆ are not mutually exclusive. The events π΄ and π΅ are mutually
exclusive, whereas events π΄ and πΆ and π΅ and πΆ are not mutually exclusive.

In our next example, we will use
the mutual exclusivity of two events and their probabilities to determine the
probability of either event occurring.

Two mutually exclusive events π΄
and π΅ have probabilities. The probability of π΄ equals
one-tenth, and the probability of π΅ equals one-fifth. Find the probability of π΄ union
π΅.

We begin by recalling that two
events π΄ and π΅ are mutually exclusive if they cannot occur at the same time. This means that there are no
elements in event π΄ and event π΅, and the probability of π΄ intersection π΅ equals
zero. The addition rule of probability
tells us that when π΄ and π΅ are mutually exclusive, the probability of π΄ union π΅
is equal to the probability of π΄ plus the probability of π΅. Mutually exclusive events can also
be represented on a Venn diagram as shown, where there is no overlap between the
circles representing events π΄ and π΅.

We are told that the probability of
event is one-tenth and the probability of event π΅ is one-fifth. The probability of π΄ union π΅ is
therefore equal to one-tenth plus one-fifth, which can be rewritten as one-tenth
plus two-tenths, which in turn is equal to three-tenths. The probability of π΄ union π΅ is
three-tenths.

In our next example, we will use
the mutual exclusivity of three events and properties of probability to determine
the probability of a compound event.

A bag contains red, blue, and green
balls, and one is to be selected without looking. The probability that the chosen
ball is red is equal to seven times the probability that the chosen ball is
blue. The probability that the chosen
ball is blue is the same as the probability that the chosen ball is green. Find the probability that the
chosen ball is red or green.

We will begin by naming the events
of choosing a red ball, a blue ball, and a green ball as R, B, and G,
respectively. Since the selected ball can only be
one of these three colors, we can conclude that the events are mutually
exclusive. Our aim in this question is to find
the probability that the chosen ball is red or green. This is the probability of the
union of the two events. And we recall for any two mutually
exclusive events π₯ and π¦, the probability of π₯ union π¦ is equal to the
probability of π₯ plus the probability of π¦. This means that we need to find the
sum of the probability that the chosen ball is red and the probability that the
chosen ball is green.

We are told that the probability
that the chosen ball is red is seven times the probability that the chosen ball is
blue. This can be written as shown. We are also told that the
probability that the chosen ball is blue is the same as the probability that the
chosen ball is green.

Finally, since there are only red,
blue, and green balls in the bag and these are mutually exclusive events, we have
the probability of red plus the probability of blue plus the probability of green is
equal to one. Replacing the probability of red
and the probability of green using equations one and two, we have the following
equation. Seven multiplied by the probability
of blue plus the probability of blue plus the probability of blue is equal to
one. This simplifies to nine multiplied
by the probability of blue is equal to one. And the probability that the
selected ball is blue is therefore equal to one-ninth.

This means that the probability
that the chosen ball is green is also equal to one-ninth. And the probability that the chosen
ball is red is seven-ninths. After clearing some space, we can
now calculate the probability that the chosen ball is red or green. This is equal to seven-ninths plus
one-ninth, which in turn is equal to eight-ninths. When a ball is selected from the
bag without looking, the probability that the chosen ball is red or green is
eight-ninths.

In our final example, we will see
two events that are not mutually exclusive.

The probability that a student
passes their physics exam is 0.71. The probability that they pass
their mathematics exam is 0.81. The probability that they pass both
exams is 0.68. What is the probability that the
student only passes their mathematics exam?

We will begin by naming the events
of passing examinations in physics and mathematics as P and M, respectively. We are told that the probability of
a student passing the physics exam is 0.71. And the probability of passing the
mathematics exam is 0.81. Since it is possible to pass both
exams, the two events are not mutually exclusive,. And we are told that the
probability of passing both exams is 0.68.

This information can be represented
on a Venn diagram, where the overlap or intersection of the two circles represents
the probability that a student passes both exams. As already mentioned, this is equal
to 0.68. The question asked us to calculate
the probability that the student only passes their mathematics exam. This will be equal to the
probability that they pass mathematics minus the probability that they pass both
exams. We need to subtract 0.68 from
0.81. This is equal to 0.13. The probability that the student
only passes mathematics, in other words, that they pass mathematics and not physics,
is 0.13.

We can add this onto our Venn
diagram, as shown. And we could repeat this process to
find the probability that the student only passes their physics exam. Subtracting 0.68 from 0.71 gives us
0.03. There is one option left in order
to complete the Venn diagram: the probability that the student doesnβt pass physics
or mathematics, in other words, that they fail both exams. 0.03 plus 0.68 plus 0.13 is equal
to 0.84. Since probabilities must sum to one
and one minus this is equal to 0.16, the probability that a student fails physics
and mathematics is 0.16.

We now have the completed Venn
diagram, which shows that the probability that the student only passes their
mathematics exam is 0.13.

We will now summarize the key
points from this video. Two events π΄ and π΅ are mutually
exclusive if the intersection of π΄ and π΅ is equal to the empty set. This means that mutually exclusive
events cannot occur at the same time since the probability of π΄ intersection π΅ is
equal to the probability of the empty set, which is equal to zero. A list of events is mutually
exclusive if they are pairwise mutually exclusive. Finally, if events π΄ and π΅ are
mutually exclusive, then the probability of π΄ union π΅ is equal to the probability
of π΄ plus the probability of π΅.