### Video Transcript

In this video, we will learn how to
interpret a data set by finding and evaluating theoretical probabilities. We begin by recalling that the
probability of an event is the likelihood of it happening. And as such, the higher the
probability of an event, the more likely it is to occur. All probabilities have values in
the range from zero to one. For example, an event with a
probability of zero can never occur and is therefore impossible, whereas an event
with probability equal to one is certain to happen.

As we can see from the number line,
we can express probabilities as both fractions and decimals, though in many examples
weβll be asked to express the probability in a certain form. When working with probability, we
often see activities such as rolling a die, tossing a coin, or selecting a ball from
a bag. For example, we may be asked to
find the probability of rolling a four on a die or the probability of selecting a
blue ball from a bag.

Since these are practical
activities and are referred to as experiments, we might think we have to carry them
out in real life in order to calculate any probabilities. However, this is not necessarily
the case. Theoretical probability describes
the behavior we expect to happen in theory. To calculate the theoretical
probability of an event, we need to be given a precise description of the system
that gives rise to it. We then use logic to analyze the
system and work out how it should behave in theory.

Before looking at some examples,
weβll first recall some basic terminology relating to experiments. The possible results of an
experiment are known as outcomes. The sample space π is the set of
all possible outcomes. An event πΈ is a set of outcomes to
which a probability is assigned. It is a subset of the sample space
π written as shown. And we use the term βfavorable
outcomesβ to refer to the outcomes we want to test for in an experiment. For a given experiment, once we
have identified the sample space and the event we want to test for, we can calculate
the theoretical probability of that event using the following formula. The probability of an event is
equal to the number of favorable outcomes divided by the total number of possible
outcomes.

More formally, this can be written
as follows. Let π be the sample space; πΈ,
which is a subset of π, be the event; π of π be the number of elements in π; and
π of πΈ be the number of elements in πΈ. Then, the probability of the event
π of πΈ is given by the formula π of πΈ is equal to π of πΈ over π of π.

We will now consider some examples
where we need to calculate the theoretical probability of an event.

A box contains five red marbles,
eight green marbles, and four yellow marbles. If a marble is picked from the box
at random, what is the probability that the marble is red?

We begin by considering the
information given in the question. We have a box containing red,
green, and yellow marbles. There are five red marbles, eight
green marbles, and four yellow marbles. And we are asked to find the
probability that a randomly selected marble is red. This is an example of theoretical
probability. And we recall that the probability
of an event can be written as a fraction where the numerator is the number of
favorable outcomes and the denominator is the total number of possible outcomes. In this question, there are five
red marbles. Since five plus eight plus four is
equal to 17, there are 17 marbles in total. And we can therefore conclude that
the probability of selecting a red marble is five out of 17 or five
seventeenths.

In our next example, we need to
find the probability of an event in a dice experiment.

What is the probability of rolling
a number greater than or equal to four on a fair die?

We begin by recalling that a
regular fair die has six faces numbered from one to six. This means that when we roll the
die, there are six possible outcomes. In this question, we are interested
in the event of rolling a number greater than or equal to four. There are three ways of satisfying
this, rolling a four, rolling a five, or rolling a six. We recall that we can write the
probability of an event as a fraction, where the numerator is the number of
favorable outcomes and the denominator the total number of possible outcomes.

In this question, the probability
of rolling a number greater than or equal to four is three out of six or
three-sixths. Since the numerator and denominator
are divisible by three, we can simplify this fraction. And we can therefore conclude that
the probability of rolling a number greater than or equal to four on a fair die is
one-half. It is worth noting that we could
also write our answer as a decimal or percentage. One-half is equal to 0.5 and 50
percent.

In our next example, weβre asked to
calculate the probability of an event without knowing the size of the sample
space.

A bag contains an unknown number of
balls. Given that one-sixth of the balls
are white, one-fifth of them are green, and the rest are blue, what is the
probability that a ball drawn at random from the bag is blue?

In this question, we are told that
a bag contains three different colored balls. They are white, green, and
blue. We are not told the number of balls
that are each color or the total number of balls in the bag. We are told, however, that
one-sixth of the balls are white. And this means that the probability
of selecting a white ball at random is one-sixth. Likewise, we are told that
one-fifth of the balls are green, so the probability of selecting a green ball is
one-fifth. It is the probability of selecting
a blue ball, that we will call π₯, that we are trying to calculate.

We recall that the sum of the
probabilities of all outcomes in a sample space must equal one. This means that in this question
the probability of selecting a white ball plus the probability of selecting a green
ball plus the probability of selecting a blue ball must equal one. Substituting in the values we know,
we have one-sixth plus one-fifth plus π₯ is equal to one. On the left-hand side of our
equation, we can begin by adding one-sixth and one-fifth by finding the lowest
common denominator, which is equal to 30. One-sixth is equivalent to five
thirtieths and one-fifth is equivalent to six thirtieths.

Adding the numerators, we see that
one-sixth plus one-fifth is equal to eleven thirtieths. Our equation simplifies to eleven
thirtieths plus π₯ is equal to one. Noting that one whole one is equal
to thirty thertieths. We can subtract eleven thirtieths
from both sides. This gives us π₯ is equal to
nineteen thirtieths. And we can therefore conclude that
the probability of selecting a blue ball from the bag is nineteen thirtieths.

In some questions, rather than
calculating probabilities of events, we are given the probability and have to work
backwards using the formula to find the size of the set of outcomes corresponding to
a particular event. This is true of our next
example.

A bag contains 30 colored
marbles. The probability of choosing a white
marble at random is two-fifths. How many white marbles are in the
bag?

We begin by recalling that the
probability of an event is equal to the number of favorable outcomes divided by the
total number of possible outcomes. This can be written more formally
as shown. π of πΈ is equal to π of πΈ over
π of π, where π is a sample space and π of π is its size, πΈ is the event weβre
interested in, and π of πΈ is its size and π of πΈ is its theoretical
probability.

In this question, the probability
of selecting a white marble is equal to the number of white marbles divided by the
total number of marbles. We are told in the question that
there are 30 marbles in total, and two-fifths of these are white. Letting the number of white marbles
be π₯, we have two-fifths is equal to π₯ over 30. We can multiply both sides of this
equation by 30 such that π₯ is equal to two-fifths multiplied by 30 or two-fifths of
30. As one-fifth of 30 is equal to six,
two-fifths of 30 is 12. We can therefore conclude that
there are 12 white marbles in the bag.

An alternative method would be to
notice that the fractions two-fifths and π₯ over 30 must be equivalent. And since five multiplied by six is
equal to 30 and two multiplied by six is equal to 12, the number of white marbles
must be equal to 12.

We will now consider one final
example involving theoretical probability.

In a class of 50 students, 33
passed the mathematics test and 31 passed the language test. What is the probability that a
randomly selected student failed in the language test?

In questions of this type, it is
important that weβre able to select the relevant information. We are only interested in what
happened in the language test. This means that the fact we are
told that 33 students passed the mathematics test is irrelevant. We are told that there are 50
students in total. Of these, 31 passed the language
test. Since 50 minus 31 is equal to 19,
we know that 19 students failed the language test. And it is the probability of this
event that we are trying to calculate.

We know that the probability of an
event can be written as a fraction. It is the number of favorable
outcomes divided by the total number of possible outcomes. The probability that a randomly
selected student failed in the language test is therefore equal to 19 over 50, which
could also be written as the decimal 0.38 or 38 percent.

We will now finish this video by
summarizing the key points.

When given a description of an
experiment, we need to identify its sample space β that is, the set of all possible
outcomes β and the event we want to measure, the set of favorable outcomes. The theoretical probability of such
an event is given by the following formula. The probability of an event is
equal to the number of favorable outcomes divided by the total number of possible
outcomes. More formally, this can be written
as shown, where π is the sample space; πΈ, which is a subset of π, is the event;
π of π is the number of elements in π; π of πΈ the number of elements in πΈ; and
π of πΈ is the probability of the event πΈ.

We saw in this video that we can
work backward from a given probability using the formula to find the size of the set
of outcomes corresponding to a particular event. Finally, we saw that we can
sometimes calculate a missing theoretical probability even if we do not know the
size of the sample space. This is because the theoretical
probability of an event is also the proportion of the total number of outcomes that
are favorable.