### Video Transcript

In this video, we will learn how to
find the probability of a simple event. We will begin by recapping some of
the key terminology associated with probability. Firstly, we recall that an
experiment is an activity with an identifiable result. An outcome is a specific result of
such an experiment. A sample space is the set of all
possible outcomes from a random experiment. Finally, an event is a subset of
the sample space where an event may consist of a single outcome or maybe composed of
multiple outcomes.

Letβs consider the experiment of
rolling a fair six-sided die and recording the number. In this case, a specific outcome
would be rolling a particular number, for example, four. The sample space would be the set
of all outcomes, in this case one, two, three, four, five, and six. And an event would be the subset of
the sample space, for example, rolling a one or rolling a number greater than
three. Informally, we may think of an
event as something that happens and the probability of an event is how likely it is
to happen. In everyday life, we may discuss
probabilities using words like βcertain,β βimpossible,β βunlikely,β or βvery
likely.β For example, we may say it is
likely that it will rain tomorrow.

In mathematics, we want to extend
this idea and start describing probabilities using numbers. We begin by defining impossible
events as having a probability of zero and events that are certain to happen as
having a probability of one. This leads to the probability scale
on which the probabilities of all events are placed between these values. Events that have a probability of
0.5, or one-half, have an even chance of occurring. This is because they are just as
likely to happen as they are to not happen. Events with a probability between
zero and 0.5 are less likely to occur. And events with a probability
between 0.5 and one are more likely to occur. We may also use terms like highly
unlikely or highly likely to describe events for which the probabilities are close
to the extremes of zero and one.

We note that probabilities can be
expressed as either fractions, decimals, or percentages. When all the outcomes of an
experiment are equally likely, the probability of an event can be calculated by
dividing the number of outcomes in that event, which we call the number of
successful outcomes by the total number of outcomes in the sample space. This can be more formally defined
as follows. If π΄ is an event in a sample space
π where each outcome is equally likely, then the probability of event π΄ occurring
is π of π΄ which is equal to π of π΄ over π of π, where π of π΄ represents the
probability of event π΄. π of π΄ represents the number of
elements in event π΄. And π of π represents the number
of elements in the sample space π.

We will now consider a series of
examples. In each example, the key will be to
determine the total number of outcomes or elements in the sample space of the
experiment and the number of outcomes which are considered to be successful.

If I roll a regular six-sided die,
what is the probability that the score is three?

As the die is regular, this means
it is unbiased, and so every outcome is equally likely. We can therefore calculate the
required probability by recalling that the probability of an event is equal to the
number of successful outcomes divided by the total number of outcomes. In this case, the event weβre
interested in is getting a three when a regular six-sided die is rolled. The possible outcomes when rolling
a die are the numbers one, two, three, four, five, and six. This means that the total number of
outcomes is six. The only successful outcome is
getting the number three. So there is one successful
outcome. We can therefore conclude that the
probability of rolling a three, written π of three, is one out of six, or
one-sixth.

In our next example, we will
calculate the probability of a simple event, which consists of more than one
outcome.

If I roll a regular six-sided die,
what is the probability that the score is divisible by three?

In this question, weβre rolling a
regular six-sided die, which means that every outcome is equally likely. There are six possible outcomes:
the numbers one, two, three, four, five, and six. And we are asked to calculate the
probability that the score that we land on is divisible by three. We recall that the probability of
an event can be written as a fraction, where the numerator is the number of
successful outcomes and the denominator is the total number of outcomes. Applying the formula to this
question, we have the probability that the score is divisible by three is equal to
the number of outcomes divisible by three divided by the total number of
outcomes. The numbers three and six are both
divisible by three. Therefore, there are two successful
outcomes.

The probability that the score is
divisible by three is therefore equal to two out of six, or two-sixths. And as both the numerator and
denominator are divisible by two, this simplifies to one-third. When rolling a regular six-sided
die, the probability that the score is divisible by three is one-third.

It is worth noting at this stage
that if we are giving probabilities as fractions, it is generally good practice to
give them in their simplest form. Letβs now consider an example in a
different context.

A card is drawn at random from a
deck of cards numbered one to 52. What is the probability that the
card drawn is a prime number?

We begin by noting that as the card
is to be drawn at random, every card has an equal chance of being chosen. In order to calculate the required
probability, we can use the formula π of π΄ is equal to π of π΄ divided by π of
π, where π of π΄ represents the probability of event π΄, π of π΄ represents the
number of outcomes in event π΄, and π of π represents the number of elements in
the sample space π. Since there are 52 cards in the
deck, π of π is equal to 52. The event weβre interested in in
this question is drawing a prime number. This means that π of π΄ will be
the number of prime number cards in the deck.

We recall that a prime number has
exactly two factors, the number one and itself. The prime numbers less than 20 are
as shown, as all of these numbers are only divisible by one and the number
itself. Continuing this list, the numbers
23, 29, 31, 37, 41, 43, and 47 are also prime. In total, we have 15 prime numbers
between one and 52 inclusive. We can therefore conclude that the
probability of event π΄ drawing a prime number is 15 out of 52. This fraction cannot be simplified
as 15 and 52 have no common factors apart from one.

In our next example, we will
calculate an experimental probability from survey data that has been presented in a
frequency table.

The table shows the results of a
survey that asked 100 people to vote for their favorite type of TV program. What is the probability that a
randomly selected person prefers drama?

The table tells us that 14 of the
surveyed people prefer drama programs. We are also told that 19 prefer
documentaries, 14 prefer comedy programs, 16 prefer news programs, and 37 prefer
sport programs. As mentioned in the question, there
were a total of 100 people surveyed. A person is to be selected at
random, and as such each person has an equal chance of being chosen. We can therefore calculate the
required probability using the formula the probability of an event is equal to the
number of successful outcomes over the number of possible outcomes. In this question, the number of
successful outcomes will be equal to the number of people who prefer drama. And the number of possible outcomes
will be the total number of people.

As already mentioned, we can see
from the table that 14 people prefer drama and there were a total of 100 people
surveyed. The probability that a randomly
selected person prefers drama is therefore equal to 14 out of 100. Dividing both the numerator and
denominator by two, this fraction simplifies to seven over 50, or seven
fiftieths. We could also write this answer as
the decimal 0.14 or 14 percent. This is the probability that a
randomly selected person prefers drama.

It is worth noting at this point
that we can use probabilities to make conclusions based on statistical data. For example, in this question, we
have found that the probability that a randomly selected person prefers drama was 14
percent. The probability that a randomly
selected person prefers sport, on the other hand, is 37 out of 100, or 37
percent. Based on this information, the
person responsible for program scheduling may choose to show a greater proportion of
sports programs than drama programs in order to appeal to a larger number of
people. It is therefore important that
whenever samples are taken, they are unbiased and representative of the population
being sampled so that any decisions made are based on reliable data.

In our final example, we will work
backwards from knowing a probability to determine an unknown.

A bag contains 16 white balls and
an unknown number of red balls. The probability of choosing a red
ball at random is one-third. How many balls are in the bag?

There are many ways of approaching
this problem. We will begin by demonstrating an
algebraic method in which we form and solve an equation. In order to determine the total
number of balls in the bag, we will first need to calculate the number of red
balls. And since this is currently
unknown, we will let the number of red balls be π. Since there are 16 white balls and
the bag only contains red and white balls, the total number of balls is equal to π
plus 16. We are also told in the question
that the probability of choosing a red ball at random is one-third. Next, we recall that the
probability of an event is equal to the number of successful outcomes divided by the
total number of outcomes. In this question, the probability
of selecting a red ball is equal to the number of red balls divided by the total
number of balls.

Substituting the information we
know into this formula, we have one-third is equal to π over π plus 16. Cross multiplying, we can rewrite
this equation as π plus 16 is equal to three π. We can then subtract π from both
sides such that two π is equal to 16. And dividing through by two, we
have π is equal to eight. This means that there are eight red
balls in the bag. And since there are 16 white balls,
there are 24 balls in total in the bag. An alternative method here would be
to notice that since one-third of the balls are red, two-thirds must be white. And since 16 balls are white, half
of this, that is, eight of the balls, must be red. This once again confirms that there
are 24 balls in total in the bag.

We will now finish this video by
recapping the key points. When all the outcomes of an
experiment are equally likely, the probability of an event can be calculated using
the following formula. The probability of an event is
equal to the number of successful outcomes over the total number of outcomes. More formally, this can be written
as π of π΄ is equal to π of π΄ over π of π, where π of π΄ represents the
probability of event π΄. π of π΄ represents the number of
outcomes in event π΄. And π of π represents the number
of elements in the sample space π.