### Video Transcript

In this video, we’ll learn how to
find the probability of a simple event. We’ll begin by defining what
probability is. And then we’ll see how we can move
from using words like “likely” or “certain” to describe the probability of an event
to using a number to describe its probability.

Probability is the likelihood or
chance of an event occurring. We often see probabilities
represented as a scale. And we use words like “impossible”
and “certain” to describe these probabilities. The number that we used to
represent an impossible event is zero. And for a certain event, it’s the
number one. Events which are equally likely as
unlikely can be described in numbers as a fraction, decimal, percentage as one-half,
0.5, or 50 percent. We can describe any probability
using a fraction, decimal, or percentage.

In order to work out the
probability of an event occurring, we work out the number of favorable outcomes and
divide it by the total number of outcomes. Let’s take the example of rolling a
die that has the numbers one to six on it. Let’s say we wanted to calculate
the probability of rolling a two. We could write this as 𝑃 brackets
two. The number of favorable outcomes
would be the number of twos on the die and the total number of outcomes would be
six, since there’s six different values on the die. And so the probability of rolling a
two on this die is one-sixth.

We can also define the probability
of an event a little more formally. If 𝐴 is an event in a sample space
𝑆, then the probability of event 𝐴 occurring is the probability of 𝐴 is equal to
𝑛 brackets 𝐴 over 𝑛 brackets 𝑆, where 𝑛 brackets 𝐴 represents the number of
elements in event 𝐴 and 𝑛 brackets 𝑆 represents the number of elements in the
sample space 𝑆. This might all sound a lot more
complicated, but let’s return to the example of rolling the die. When we consider the probability of
rolling a two, the value on the numerator is the number of elements in event 𝐴. So in the case of the die, that’s
how many values there are that are the number two. Then on the denominator, it’s the
number of elements that there are on the die in total. And that’s six, since there are six
different numbers. And so, the probability of rolling
a two is one-sixth.

Before we look at some examples,
there’s one very important thing to remember about probability. And we’ve already seen this when we
looked at the number line. The probability of an event must be
between zero and one. For an event 𝐴, we can say that
zero is less than or equal to the probability of 𝐴 is less than or equal to
one. In other words, the probability of
an event must be between zero and one, although of course it may be exactly equal to
zero or exactly equal to one. Let’s now take a look at some
examples. And in the first example, we’ll
find the probability of an event where there are three possible outcomes.

A bag contains seven white marbles,
eight black marbles, and seven red marbles. If a marble is chosen at random
from the bag, what is the probability that it is white?

We can recall that to find the
probability of an event occurring, we take the number of favorable outcomes and
divide it by the total number of outcomes. Here, we’re asked to calculate the
probability of getting a white marble. And we can write this as 𝑃
brackets 𝑊 for white. We then need to establish how many
white marbles there are. And we’re told that there are
seven. So this will be the value on the
numerator. The value on the denominator will
be the total number of marbles. So we’re told that there are seven
white, eight black, and seven red. So we add these values
together. This gives us the fraction seven
over 22. We can’t simplify this fraction any
further. So the probability of getting a
white marble is seven over 22.

We’ll now look at another example
of calculating the probability of an event.

What is the probability of
selecting at random a prime number from the numbers eight, nine, 20, 19, three, and
15?

In order to answer this question,
we’ll need to recall two things, firstly, how we find the probability of an event,
and, secondly, what a prime number is. In order to calculate the
probability, then we can use the formula to calculate the probability of an event 𝐴
as the number of elements in event 𝐴 divided by the number of elements in a sample
space 𝑆. We may also see this formula
written as the probability of an event is equal to the number of favorable outcomes
over the total number of outcomes. Here, we’re trying to calculate the
probability of selecting a prime number. A prime number is a number that has
exactly two factors, one and itself.

So let’s consider the list of
numbers that we were given. The first three values eight, nine,
and 20 are not prime because they have more than two factors. Next, 19 and three are both prime
numbers. However, 15 is not a prime
number. When it comes to using the formula
then, the number of prime numbers that we have is two. That’s three and 19. The value on the denominator will
be six, since there were six numbers in total. It’s always good to simplify our
fractions where we can. And so we can give the answer that
the probability of selecting a prime number from the given list of numbers is
one-third.

In the next example, we’ll see how
we can find the number of elements in the sample space, given the number of elements
in the event and the probability of a different event.

A bag contains 24 white balls and
an unknown number of red balls. The probability of choosing at
random a red ball is seven over 31. How many balls are there in the
bag?

Because we’re calculating a
probability, we can recall that to find the probability of an event 𝐴, we calculate
𝑛 brackets 𝐴 divided by 𝑛 brackets 𝑆, where 𝑛 brackets 𝐴 is the number of
elements in event 𝐴 and 𝑛 brackets 𝑆 is the number of elements in sample space
𝑆. Let’s have a closer look at this
question then. We’re told that there are 24 white
balls in the bag, but an unknown number of red balls. We can use the same style of
notation then to note that the number of white balls must be 24. We’re not told the number of red
balls, but we can use a variable such as 𝑥 to represent this. The total number of balls in the
bag can be written as the number of white balls, that’s 24, plus the number of red
balls, which we’ve defined as 𝑥. The total number of balls in the
bag will be the number of elements in the sample space. So we can say that 𝑛 brackets 𝑆
is equal to 24 plus 𝑥.

Next, we can have a look at the
probabilities. And we’re told that the probability
of picking a red ball is seven over 31. We can now write this probability
formula in terms of finding the probability of a red ball. Then we’ll see if we have enough
information to solve to find the value of 𝑥. To calculate the probability of
getting a red ball, well, the number of elements in any event 𝐴 for a red ball
would correspond to the number of red balls in the bag. The denominator, which is the
number of elements in sample space 𝑆 in this context, is simply the total number of
balls in the bag. We can then plug in the values that
we have for these expressions. So we have seven over 31 is equal
to 𝑥 over 24 plus 𝑥.

Now, we can take the cross product
and solve this for 𝑥. We then distribute seven across the
parentheses, which gives us 168 plus seven 𝑥 is equal to 31𝑥. Then we can subtract seven 𝑥 from
both sides. Finally, dividing both sides by 24,
we find that seven is equal to 𝑥, and so 𝑥 is equal to seven. Remember that we defined the number
of red balls to be 𝑥, and so we’ve worked out that the number of red balls must be
equal to seven. It can be very tempting to stop
here. But remember, we’re asked how many
balls there are in total in the bag. We’ve already worked out an
expression for the number of balls in the bag. It was that given by this number of
elements in the sample space. So the total number of balls is 24
plus 𝑥, which was seven. And when we add those together, we
get the value of 31. Therefore, we can give the answer
that there must be 31 balls in the bag.

In the final example, we’ll see how
we can find the total number of elements in a sample space given the probabilities
of two events and the number of occurrences of a third event.

A bag contains an unknown number of
marbles. There are three red marbles, some
white marbles, and some black marbles. The probability of getting a white
marble is one-third and the probability of getting a black marble is one-half. Calculate the number of marbles in
the bag.

Because we have a probability
question here, we can recall that to find the probability of an event 𝐴, we
calculate the number of elements in event 𝐴 which we can write as 𝑛 brackets 𝐴
and we divide by the number of elements in the sample space 𝑆 which we can write as
𝑛 brackets 𝑆. If we have a look at the question,
we’re told that there are three different colors of marbles in the bag: red, white,
and black. So let’s jot down some information
from the question. We can use the same notation that
the number of red marbles must be 𝑛 brackets 𝑅, which we could say is equal to
three.

We aren’t given the number of white
marbles or black marbles, but let’s define the number of white marbles as 𝑤 and the
number of black marbles as 𝑏. Knowing these three values or
expressions for these values will allow us to write down that the total number of
marbles in the bag, which is the same as the number of elements in the sample space,
the number of marbles in the bag. That’s three plus 𝑤 plus 𝑏. Next, let’s look at the
probabilities. We’re told that the probability of
getting a white marble is one-third and the probability of getting a black marble is
one-half.

Now, if we look at the information
that we’ve jotted down, we can see that we have two unknown quantities, 𝑤 and 𝑏,
which is the number of white marbles and the number of black marbles. What we’ll aim to do is to get
enough information to define two equations to solve for these two unknowns. We can write the general
probability formula in terms of finding the probability of a white marble. In this case, it will be equal to
the number of white marbles over the total number of marbles.

Since we were given that the
probability of getting a white marble is one-third, we have that on the left-hand
side and then, on the right-hand side, the number of white marbles was equal to
𝑤. And the total number of marbles is
equal to three plus 𝑤 plus 𝑏. Taking the cross product, we have
three plus 𝑤 plus 𝑏 is equal to three 𝑤. We can then subtract the 𝑤 from
both sides to give us that three plus 𝑏 is equal to two 𝑤. We can’t do anything more with this
equation at the minute, but let’s label it equation one. And we’ll move it off to one side
to clear some space. And hopefully, we can get a second
equation in terms of 𝑤 and 𝑏, which we can then solve simultaneously to find 𝑏
and 𝑤.

What we can do next, then, is to
write the probability formula in terms of finding the probability of getting a black
marble. This time, we’ll be calculating the
number of black marbles over the total number of marbles. And so, on the left-hand side,
we’ll have one-half since that’s the probability of getting a black marble. And on the right-hand side, we’ll
have 𝑏 divided by three plus 𝑤 plus 𝑏. We can take the cross product to
give us three plus 𝑤 plus 𝑏 is equal to two 𝑏. Subtracting 𝑏 from both sides
gives us three plus 𝑤 is equal to 𝑏. And so we have found a second
equation in terms of 𝑤 and 𝑏.

If we look at these two equations,
there are a number of different ways in which we could solve these. But if we look at the second
equation, we have an expression for 𝑏. So we can substitute this into
equation one in place of 𝑏. This gives us three plus three plus
𝑤 is equal to two 𝑤. We can then simplify this and
subtract 𝑤 from both sides, which gives us that six is equal to 𝑤. This means that the number of white
marbles is six.

To find 𝑏, the number of black
marbles, we can substitute 𝑤 equals six into either equation, but let’s plug it
into equation two. This gives us three plus six is
equal to 𝑏. And therefore nine is equal to 𝑏,
so 𝑏 is equal to nine. The number of black marbles is
nine. To find the total number of marbles
in the bag, we already have an expression for it. It’s three plus 𝑤 plus 𝑏. So when we plug in the values 𝑤
equals six and 𝑏 equals nine, we have three plus six plus nine. Working this out, we have the
answer that the number of marbles in the bag must be 18.

And, of course, it’s always
worthwhile checking our answers. We were given that there are three
red marbles. We worked out that there must be
six white marbles and nine black marbles. And those do indeed add up to
18. In terms of the probabilities, we
were told that the probability of white is one-third and the probability of black is
one-half. We could work out the probability
of getting a red as three out of 18, which simplifies to one-sixth. When we add the three probabilities
of one-sixth, one-third, and one-half, we get a total of one. Since the sum of the probabilities
must be one, then this will confirm that the values that we worked out to get a
total number of marbles as 18 must be correct.

We can now summarize the key points
of this video. We saw that there are two formulas
that we can use to calculate the probability of an event. The second formula is particularly
useful when we’re dealing with the probabilities of a number of events, as we saw in
a question like the previous one. And finally, we saw that all
probabilities must lie in the interval zero, one. A probability that’s closer to zero
would indicate an event that is less likely, and a probability that’s closer to one
would indicate an event which is more likely.