Video Transcript
In this video, we will learn how to
identify mutually exclusive events and then find their probability. If two or more events are mutually
exclusive, they cannot happen at the same time. For example, an animal cannot be a
cat and a dog, which means being a dog is mutually exclusive with being a cat. And while being a dog and being a
cat are mutually exclusive, liking dogs is not mutually exclusive to liking
cats. There might be some people who like
dogs, some people who like cats, and some people who like both dogs and cats, which
means this category is not mutually exclusive. Because both things can be true at
the same time. Because there is no overlap between
dogs and cats, we call these categories mutually exclusive. You might also hear it called
“disjoint.”
Before we look at finding the
probability of mutually exclusive events, let’s quickly remind ourselves of some
rules of probability. For any event A, if the probability
of A is the probability of A occurring, the following is true. The probability of A must be
between zero and one. The sum of the probabilities of all
possible outcomes is equal to one, that is, 100 percent. The compliment of event A written
as A with a bar over it refers to everything that is not A. And the probability of the
compliment of A is equal to one minus the probability of A. You might occasionally see the
compliment be written as A prime, A with a dash, as well.
But now we want to look at finding
the probability of mutually exclusive events and non-mutually exclusive events. Let’s say we want to find the
probability that event A or event B happens. We write that, the probability of A
or B, mathematically like this. On the left, we have mutually
exclusive events. And we want to know the probability
of A or B happening. And that would be the whole
probability in the blue, the probability of A, plus the whole probability of B, the
probability of the yellow.
For example, if you had the
probability that someone would choose a dog for a pet and the probability that
someone would choose a cat for a pet. The probability of A or B would be
the probability that they would choose a dog or a cat. And to find that, we would need to
add these two probabilities together.
Let’s try the same thing with our
non-mutually exclusive event. The probability of event A is
everything in the blue circle. If we add the probability of event
B to that — that’s the whole yellow circle — the problem is that, in non-mutually
exclusive events, A and B share a probability. They share the probability of A and
B. That’s that intersection
probability.
To find the probability of
non-mutually exclusive events, we take the probability of event A plus the
probability of event B. And then we subtract the overlap
between them.
Going back to our example from
earlier, if we let A be the probability that someone likes dogs, B be the
probability that someone likes cats, then the probability of A and B is the people
who like both. And that means the probability that
people like cats or dogs is equal to the probability that people like dogs includes
the people who like both cats and dogs. And the probability that someone
likes cats includes the people who like both cats and dogs. And that means we’ve added this
group twice.
So if we subtract the people who
like both cats and dogs, it cancels out that repeat intersection. So that we have the people that
like dogs — and that includes the people that like cats and dogs — plus the people
that like only cats. In mutually exclusive events, we
don’t have to do this because the probability of A and B equals zero. There’s no overlap. So to find the probability of A or
B in mutually exclusive events, we add the probability of A to the probability of B
because the probability of A and B is zero. Now we’re ready to look at some
examples.
In an animal rescue shelter, 39
percent of the current inhabitants are cats, C, and 41 percent are dogs, D. Find the probability that an
animal chosen at random is either a cat or a dog. Find the probability that an
animal chosen at random is neither a cat nor a dog.
Let’s see what we know. In an animal rescue shelter, 39
percent of the animals are cats and 41 percent of the animals are dogs. For problem one, we wanna find
the probability that an animal chosen at random is either a cat or a dog.
We know that animals can either
be a cat or a dog, but not both, which makes these mutually exclusive
events. And that means, to find the
probability of event C or D happening, we add the probability of event C and the
probability of event D. Remember, the probability of
event A occurring is the number of ways A can occur over all possible
outcomes. Probability of C would be the
probability a cat is chosen out of all possible animals.
If 39 percent of all the
animals are cats, then there is a probability of 0.39 that a cat would be chosen
out of all possible animals. In the same way, if 41 percent
of all the animals are dogs, then the probability of choosing a dog out of all
possible animals is 0.41. If we combine those two
probabilities, 0.39 plus 0.41, we see that the probability of selecting a cat or
a dog, if chosen randomly, is 0.80. That also means we know that 80
percent of the animals in the shelter are either a cat or a dog. If 80 percent of the animals
are either cats or dogs, then 20 percent of the animals are not cats or
dogs.
To find the probability that an
animal chosen at random is neither a cat nor a dog, we could find the compliment
of the probability that it is a cat or a dog. Since the compliment is
everything that is not C or D. And we find that by taking the
probability of C or D and subtracting that from one. The probability that you will
not select a cat or a dog is then 0.20.
Let’s look at another example. This time, we only need to decide
if events are mutually exclusive or not.
Amelia has a deck of 52
cards. She randomly selects one card
and considers the following events. Event A, picking a card that is
a heart. Event B, picking a card that is
black. Event C, picking a card that is
not a spade. Are events A and B mutually
exclusive? Are events A and C mutually
exclusive? Are events B and C mutually
exclusive?
Let’s take each one of these
questions in turn, starting with the first one. Are events A and B mutually
exclusive? Event A is picking a card that
is a heart, and event B is picking a card that is black. If we consider a standard deck
of 52 cards, event A would be selecting any one of the cards that is hearts. And this is event B, picking a
card that is black.
In mutually exclusive events,
the probability of A and B is zero. It’s not possible for both
events to happen at the same time. When we’re asking, “Are A and B
mutually exclusive?,” we should ask, can A and B happen at the same time? Can Amelia choose a card that
is a heart and black? No, that is not possible. Since it’s not possible for A
and B to be true at the same time, these events are mutually exclusive.
What about events A and C? Event A is the same. Because event C involves
picking a card that is not a spade, it could be clubs, hearts, or diamonds. We ask the same question. Can event A and event C happen
at the same time? That is possible. Since they can both be true at
the same time, these events are not mutually exclusive.
What about B and C? With the same question, can B
and C happen at the same time? It is possible to choose a card
that is black and is not a spade. Those would be any cards that
are clubs. Because event B and C can
happen at the same time, they’re not mutually exclusive.
Let’s look at another example.
A small choir has a tenor singer,
three soprano singers, a baritone singer, and a mezzo-soprano singer. If one of their names was randomly
chosen, determine the probability that it was the name of the tenor singer or the
soprano singer.
The choir has one tenor, three
sopranos, one baritone, and one mezzo-soprano. If we let them be mutually
exclusive, that means the tenor would not also be the baritone. We would say there is six total
people. We want to know the probability of
tenor or soprano. And because they’re mutually
exclusive, we can just add these values together, the probability of a tenor and the
probability of a soprano.
Since there’s only one tenor, the
probability of randomly selecting that person is one out of six. And since there are three sopranos,
the probability of randomly selecting their name would be three out of six. Together, that’s a probability of
four-sixths, which can be reduced by dividing the numerator and denominator by
two. And that means the probability of
randomly selecting a tenor or a soprano is two-thirds.
Let’s look at another example.
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 want to find the probability
that we select a red or a green ball when we draw one out. We know that these events are
mutually exclusive because the ball could not be red and green at the same time. The probability of the ball being
both red and green is zero. And that means the probability that
the ball is red or green will be equal to the probability that the ball is red plus
the probability that the ball is green.
So now we need to find those
values. We know that the probability that
the ball is red is equal to seven times the probability that the ball is blue. And the probability that the ball
is green is equal to the probability that the ball is blue. We can also say that the
probability of red plus the probability of green plus the probability of blue has to
equal one.
Since the sum of the probabilities
of all possible outcomes is always equal to one. In this equation, if we substitute
the probability of blue and for the probability of green, we can substitute seven
times the probability of blue and for red. So seven times the probability of
blue plus the probability of blue plus the probability of blue has to equal one.
We can combine like terms and say
that we have nine times the probability of blue is equal to one. And that means we could divide both
sides of this by nine to show the probability of selecting blue is equal to
one-ninth. If the probability of selecting
blue is one-ninth, then the probability of selecting green is also one-ninth as they
are equal. The probability of selecting red is
equal to seven times the probability of selecting blue, which means it will be
seven-ninths. To find the probability of red and
green, then we combine seven-ninths and one-ninth to get eight-ninths.
It’s probably worth noting here
that once we found the probability of blue, we could’ve found the probability that
it is not blue. Because there’re only red, green,
and blue in the bag, the probability that it is not blue will be the same as the
probability of it being red or green, which again would be eight-ninths.
In our final example, we’re given
the probability of 𝐴 or 𝐵. And then we’re asked to find the
probability of 𝐵.
Suppose 𝐴 and 𝐵 are two mutually
exclusive events. Given that the probability of 𝐴 or
𝐵 equals 0.93 and the probability of 𝐴 not 𝐵 is equal to 0.39, find the probability
of 𝐵.
The first thing we know is the
probability of 𝐴 or 𝐵 equals 0.93. But we also know that these are
mutually exclusive events, which means the probability of 𝐴 or 𝐵 equals the
probability of 𝐴 plus the probability of 𝐵. It also means the probability of 𝐴
and 𝐵 both happening at the same time is zero. We can represent these two mutually
exclusive events with two circles that do not overlap.
We’re also given that the
probability of 𝐴 minus 𝐵 equals 0.39. This would be the probability of 𝐴
happening and not 𝐵. And that really tells us that the
probability of 𝐴 happening is 0.39. We know the probability of 𝐴 plus
the probability of 𝐵 equals 0.93. And then we plug in 0.39 for the
probability of 𝐴. To solve for 𝐵, we subtract 0.39
from both sides. And we get the probability of 𝐵 is
0.54.
Finally, let’s review our key
points. If two events cannot occur at the
same time, they are mutually exclusive, which means the probability of 𝐴 or 𝐵 is
equal to the probability of 𝐴 plus the probability of 𝐵. And it means the probability of 𝐴
and 𝐵 happening at the same time is zero.
