Video Transcript
In this lesson, we’ll learn how to
find the number of all possible outcomes in a sample space by using the fundamental
counting principle, sometimes called the product rule for counting. Let’s begin by considering an
example.
Say we have the letters A, B, and
C. How many different ways do we have
to order these?
Well, we could use something called
systematic listing; that is, we just write them all out by using some sort of
system. Let’s start with a fairly obvious
first option. Let’s start with A-B-C. Then, we’re going to keep A in
place, and we’re going to switch the other two letters to get A-C-B. That is a different order, and so
that’s the second way of ordering them. There are no other ways to order
these with A at the beginning, so we’re now going to put B at the beginning. We could have B-A-C. And if we then hold the B in place
and switch the other two letters, we get B-C-A. We can repeat this process also
keeping C at the beginning. And then, we see that we’ve run out
of ways to order this. And so, if we count these up, we
see that there are exactly six ways to order these.
And this can be a nice
technique. However, there are a few
issues. Firstly, if we have lots of
options, say we wanted to order six or seven letters, this process could take a
really long time. And secondly, it would be very easy
to miss a few options. So, let’s see if we can find
another way of doing this. Let’s look at using a tree diagram
to find the total number of possible outcomes.
If we think about the first set,
then we know that there are three different possible choices. We can choose A, B, and C. Now, let’s imagine we’ve chosen the
first letter and we’ve got A. The second letter can now only be B
or C. If, however, we’d gone down the
second path, the second letter could possibly be A or C. And if we’d gone down the third
path or the third branch and chosen a letter C , our second letter could either be A
or B. Then, when we move on to the third
letter, we see if we’ve already chosen an A and a B, we can now only choose a C. And we can repeat this by drawing
one branch off of each of the six previous branches. So, let’s count up the total number
of outcomes and see where it’s come from.
There were three branches under our
first letter. Then, under the second letter, we
had two branches coming off of each of those original three. So, there are three times two
branches. Finally, under the third letter,
there is one branch coming off of each of the previous ones. So, there’s a total of three times
two times one, which is six outcomes. And so, the number of ways of
arranging the letters here is three times two times one; it’s six. We simply multiply the number of
ways of choosing the first letter by the number of ways of choosing the second by
the number of ways of choosing the third. And, in fact, we can generalize
this.
The fundamental counting principle,
sometimes called the product rule for counting, says that if A and B are two
independent events such that A has 𝑚 outcomes and B has 𝑛 outcomes, then the total
number of distinct outcomes of the two events together is the product of these; it’s
𝑚 times 𝑛. So, let’s look at an example of the
application of this principle.
How many three-different-digit
numbers could be formed from the set of digits containing the elements one, two,
four, and nine?
Remember the fundamental
counting principle, sometimes called the product rule for counting, says that if
A is an event that has 𝑚 outcomes and B is an event that has 𝑛 outcomes, the
total number of outcomes of A and B together is the product of these. It’s 𝑚 times 𝑛. Now, in fact, we have three
possible events here. We have the event which is
choosing the first digit, the second event is choosing the second digit, and the
third event is choosing the third digit. But the counting principle
still holds, so we need to find out the number of outcomes we have for choosing
each digit and then multiply these together.
There are four possible digits
we can choose from. Those are one, two, four, and
nine. And so, it’s quite clear to us
that there are four different ways of choosing the first digit. Now, we’re told that they are
three-different-digit numbers. So, let’s think about how we
choose the second digit. Let’s say, for instance, the
first digit we chose was the number one. We can no longer use that. And so, there are three
different digits to choose from. The number of ways then of
choosing our second digit is three. Similarly, let’s move on to the
third digit. We’ve now already chosen two
numbers out of the set of four. And so, that means that there
can only be two numbers left to choose from.
And so, there are four ways of
choosing the first digit, three ways of choosing the second, and two ways of
choosing the third. The product rule for counting
or the counting principle tells us that we can find the total number of outcomes
by multiplying these together. That’s four times three times
two, which is equal to 24. There are 24
three-different-digit numbers that could be formed from the set of digits one,
two, four, and nine.
The menu at a restaurant is
shown below. How many ways are there to
choose a two-course meal?
So to choose a two-course meal,
we’re going to choose one starter and one main. And so, one way that we have to
find the total number of meals that there are is to list out all possible
options. But this can be quite a
long-winded method. And so, instead, we’re going to
recall the counting principle or the product rule for counting. This says that if A is an event
that has 𝑚 outcomes and B is an event that has 𝑛 outcomes, the total number of
outcomes for the two events together is 𝑚 times 𝑛. And this is why it’s called the
product rule for counting. We find the product of the
number of outcomes.
We see that there are four
possible ways of choosing a starter, and there are three possible ways of
choosing a main. This means then that the total
number of ways of choosing a two-course meal is the product of the number of
outcomes. It’s four times three, which is
equal to 12. There are 12 ways to choose a
two-course meal.
Let’s consider another example.
A construction company has
three active sites. There are 20 different ways to
drive from site A to site B. There are 16 ways to drive from
site B to site C. In how many ways can we drive
from site A to site C visiting site B on the way?
Let’s imagine we have these
three construction sites. We’re told that there are 20
ways to get from site A to site B and then 16 ways to get from site B to site
C. We need to work out the total
number of ways that we can get from site A to site B and then to site C. And to do so, we’re therefore
going to recall something called the fundamental counting principle. This says that if A is an event
that has 𝑚 outcomes and B is another event that has 𝑛 outcomes, then the total
number of outcomes for A and B combined is 𝑚 times 𝑛.
Our first event is driving from
site A to site B. So, there’s 20 different ways
of doing this. There’s 20 outcomes. Then, our second event is
driving from site B to site C, and we know that there are 16 outcomes. And so, the total number of
ways that we can drive from site A to site B and then to site C is 20 times 16,
which is 320. There are 320 different ways of
doing that journey.
How many different ways can we
pick a team of one man and one woman from a group of 23 men and 14 women?
This scenario effectively has
two events. Event one is choosing one man
from a total of 23, and event two is choosing one woman from a total of 14. And so, we’re going to need to
recall the fundamental counting principle. This says that if A and B are
two independent events such that A has 𝑚 outcomes and B has 𝑛 outcomes, there
are a total number of 𝑚 times 𝑛 outcomes of these two events together. And so, the total number of
ways that we can pick a team of one man and one woman must be 23 times 14. We can use any method we want
really to perform this calculation. Let’s use a column method.
Three times four is 12. So, we put a two in this column
and we carry the one. Then, two times four is eight
and we add the one to get nine. Next, we’re going to do three
times one. But since the one is in the
tens column, this is just like doing three times 10, and so we add a zero. Three times one is three and
two times one is two. We’re going to add these
values. Two add zero is two, nine add
three is 12, so we’re going to carry a one, and two add one is three. And so, we see that there are
322 ways to pick a team of one man and one woman from a group of 23 men and 14
women.
We’re going to consider just one
final example.
Two spinners are spun. The first spinner is numbered
from one to five, and the second spinner is numbered from one to seven. Determine the total number of
possible outcomes.
To answer this question, we’re
going to recall the fundamental counting principle. This says that if A and B are
independent events, that is, the outcome of one does not affect the outcome of
the other and if A has 𝑚 possible outcomes and B has 𝑛 possible outcomes, the
total number of possible outcomes of the two events together is 𝑚 times 𝑛;
it’s the product of these.
Now, our two events are
spinning the first spinner and spinning the second spinner. And so, we need to consider the
total number of outcomes for each event. The first spinner is numbered
from one to five, so there are five different scores that we can receive when we
spin that first spinner. Then the second spinner is
numbered from one to seven. So, there are seven different
outcomes; there are seven different scores that we could get. The product rule for counting
or the counting principle says that the total number of possible outcomes is the
product of these two numbers. It’s five times seven and of
course that is equal to 35. And this means that when we
spin these two spinners, we can get a total number of 35 possible outcomes.
We’ll now consider the key points
from this lesson. In this video, we learned that the
fundamental counting principle, which is sometimes called the product rule for
counting, can help us to save time when trying to find a total number of outcomes of
more than one event. This says that given two
independent events A and B, in other words, the outcome of one doesn’t affect the
outcome of the other, where A has 𝑚 outcomes and B has 𝑛 outcomes, the total
number of outcomes for these events together is the product of these numbers. It’s 𝑚 times 𝑛.
