### Video Transcript

In this video, we will learn how to
interpret a data set by finding and evaluating theoretical probabilities. We will begin by looking at the
definition of probability.

The probability of an event
occurring is the chance of it happening, for example, when rolling a die or tossing
a coin. The probability scale goes from
zero to one, where a zero probability means an event is impossible and one means
that it is certain to happen.

Probabilities can be written as
fractions, decimals, or percentages. When writing the probability as a
fraction, the numerator is the number of successful outcomes and the denominator is
the number of possible outcomes. We will now look at some questions
involving theoretical probability.

What is the probability of rolling
a number between zero and two on a fair die?

A fair die is cubic in shape and,
therefore, has six faces. Each of the faces is numbered from
one to six, and the opposite faces sum to seven. The probability of an event
occurring can be written as a fraction where the numerator is the number of
successful outcomes and the denominator the number of possible outcomes.

In this question, we want to roll a
number between zero and two. The only number that appears on a
die that is between zero and two is one. This means that there is one
successful outcome in this event. As there are six numbers in total
on the die, there are six possible outcomes. The probability of rolling a number
between zero and two on a fair die is one out of six or one-sixth.

We will know look at another word
problem involving theoretical probability.

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 are told in the question that we
have five red marbles. There are eight green marbles. And finally, we have four yellow
marbles. This means that we have 17 marbles
in total, as five plus eight plus four equals 17. We are also told that a marble is
selected at random. This means there is an equal chance
of selecting each of the 17 marbles. We know that probability can be
written as a fraction: the number of successful outcomes over the number of possible
outcomes.

In this question, we are trying to
calculate the probability that the marble is red. As there are five red marbles in
the box, the number of successful outcomes is five. The number of possible outcomes is
17 as there are 17 marbles in total. The probability of picking a red
marble from the box is five out of 17 or five seventeenths.

Our next question involves working
out the probability of an event in context.

A school club has 25 boys and 13
girls. Of the members in the club, five of
the boys and four of the girls wear glasses. If a club member is chosen at
random, what is the probability that they do not wear glasses?

There are lots of ways of
approaching this problem. One way would be to set up a
two-way table. The two-way table can be drawn as
shown. We are told in the question that
there are 25 boys in the club. There are 13 girls in the club. As 25 plus 13 is equal to 38, there
are 38 members altogether. We are also told that five of the
boys wear glasses. This means that 20 of the boys did
not wear glasses as 25 minus five is 20. Four of the girls in the club wear
glasses. This means that nine do not. The total number of students that
wear glasses is nine as five plus four equals nine. 20 plus nine is equal to 29. Therefore, there are 20 students in
the club who do not wear glasses.

We recall that probability can be
written as a fraction where the numerator is the number of successful outcomes and
the denominator the number of possible outcomes. We need to work out the probability
that a student does not wear glasses. There are 29 club members that do
not wear glasses. Therefore, our numerator will be
29. The total number of students in the
club is 38. Therefore, the denominator is
38. We can therefore conclude that the
probability that a club member does not wear glasses is 29 out of 38.

Our next question also involves
solving a real-life problem involving theoretical probability.

A bag contains white, red, and
black balls. The probability of drawing a white
ball at random is 11 out of 20 and a red ball at random is three out of 10. What is the smallest number of red
balls and black balls that could be in the bag?

We are told in the question that
there are three different color balls in the bag. The probability of selecting a
white ball is 11 out of 20 or eleven twentieths. The probability of selecting a red
ball is three out of 10 or three-tenths. We are not given the probability of
selecting a black ball.

In order to compare fractions, we
need to ensure that the denominators are the same. The lowest common multiple of 10
and 20 is 20, so we need to multiply the denominator of the second fraction by
two. Whatever we do to the denominator,
we must do to the numerator. Three multiplied by two is equal to
six, and 10 multiplied by two is 20. Therefore, the fractions
three-tenths and six twentieths are equivalent.

We know that the sum of all
probabilities is one. In this question, as our
denominator is 20, this is equal to twenty twentieths. Eleven twentieths plus six
twentieths is equal to seventeen twentieths. When the denominators are the same,
we just add the numerators. Subtracting this from one or twenty
twentieths gives us three twentieths. This means that the probability of
selecting a black ball is three twentieths.

We now have three probabilities
that we can compare as the denominators are all the same. The ratio of white to red to black
balls is 11 to 6 to three. This means that the smallest number
of balls in total is 20, where 11 would be white, six red, and three black. The smallest number of red balls
and black balls that could be in the bag are six and three, respectively.

As we’re only given the
probabilities, the total number of balls could be any multiple of 20. For example, we could have 40 balls
in total where 22 are white, 12 are red, and six are black. However, as we were looking for the
smallest number of red and black balls, the correct answer is six red and three
black.

We will now look at one final
question involving theoretical probability.

At a school fair, 68 people entered
a raffle. Given that the probability of a
girl winning the raffle is one-quarter and people can only enter once, how many
girls took part in the raffle?

We are told in the question that 68
people entered the raffle and that they can only enter once. This means that 68 tickets were
sold in total. We are also told that the
probability of a girl winning is one-quarter. This means that one-quarter of the
tickets were bought by girls. To work out the number of girls
that took part in the raffle, we need to calculate one-quarter of 68. The word “of” in mathematics means
multiply or times, so we need to multiply one-quarter by 68.

Multiplying by a quarter is the
same as dividing by four, so we can divide 68 by four. Six divided by four is equal to one
remainder two, and 28 divided by four is seven. 68 divided by four is therefore
equal to 17. We can therefore conclude that 17
girls took part in the raffle.

An alternative method to find a
quarter of a number would be to halve it and then halve it again. One-half of 68 is 34, and one-half
of this is 17.

We will now summarize the key
points from this video. To calculate the probability of a
single event, we need the following: firstly, the number of ways we can get a
successful outcome, that is, the number of ways the thing we want to happen can
happen. Secondly, we need the total number
of all possible outcomes. This means the number of things
that could possibly happen. The probability of a single event
occurring can be written as a fraction, decimal, or percentage. When written as a fraction, it is
the number of successful outcomes out of or over the number of possible
outcomes.

Finally, we recall that the
probability scale goes from zero to one. This means that the probability
written as a fraction cannot be top heavy or improper. The numerator must be less than or
equal to the denominator. If the probability of an event
occurring is zero, it is impossible. And if it is equal to one, it is
certain to happen.