Video Transcript
Experimental Probability
In this video, we will learn what
experimental probability is and try to use data sets to calculate it.
Let’s start by reminding ourselves that
probability is the likelihood or chance of an event happening. For example, we could say that the
probability of rain tomorrow is 0.1, with that the probability of rolling a five on this
spinner is one-fourth. The values that we gave for probability
will always be between zero and one and can be written as a fraction, decimal, or
percentage. In this video, we’re going to be talking
about the experimental probability of an event. This is often referred to as the relative
frequency. Here, we make estimates for the
likelihood of an event based on the results of a number of experiments.
Let’s imagine that we wanted to find the
experimental probability of a coin flip, where we get heads or tails on the coin. And we could set up a table to record the
results. For example, if we flipped the coin six
times getting four heads and two tails, we could record the results as follows. Here, the frequency is the total in each
category. And our value six would give us the total
number of trials. Continuing the experiment, we may find
that we get eight heads, 12 tails, and a total of 20 trials.
In a much longer experiment of 1000
trials, we could record the following results. If we wanted to find the probability of
getting tails on this coin, we can use the notation of 𝑃 and then tails in parentheses. Since we find the frequency of tails to
be 518, we would write this as a fraction of 518 over 1000 since our total frequency is
1000. This fraction can be further simplified
or written as a decimal, if required.
We can now see the general method of
finding the experimental probability of an event, which tells us that the relative frequency
or experimental probability of event E, where the probability of event E is equal to the
number of times that E occurs divided by the total number of trials. Before we look at some questions, let’s
have a think about why we might use experimental probability. After all, if we we’re looking at a coin
and we wanted to calculate the probability of tails, then we can use the fact that since we
just have one tails out of two possible options. Then surely the probability of tails is
equal to a half, which is in fact the theoretical probability of getting tails on a coin
flip.
So to answer the question of why we use
experimental probability, one scenario would be when we can’t calculate the theoretical
probability, for example, if we had a biased coin or awaited die. Experimental probability is widely used
in research, economics, medicine, and social sciences. In these cases, surveys or experiments
are done to calculate the probability of certain outcomes. In experimental probability, we must
ensure that we have a large enough sample size in order to give the most accurate
results.
Now, let’s take a look and work through
some questions on experimental probability.
The table shows the results of a survey
that asked 20 students about their favourite breakfast. What is the probability that a randomly
selected student prefers eggs?
In this question, to calculate the
probability, we’re going to work out the experimental probability, which is also referred
to as relative frequency. We can use the calculation that the
relative frequency of an event E is equal to the number of times E occurs divided by the
total number of trials. So, for this question, to find the
probability that a student prefers eggs, this is equivalent to finding the relative
frequency of a student who prefers eggs and will be equal to the number of students who
prefer eggs over the total number of students.
We then use the table to establish that
there are 10 students who prefer eggs and 20 students in total. Even if we hadn’t been given that the
20 students were asked, we could’ve calculated this value by adding up the values of 10,
two, and eight in the table. We can then simplify our fraction 10
over 20 as one-half. So, our final answer as a decimal for
the probability that a randomly selected student prefers eggs is 0.5. In this case, our fractions 10 over 20,
one-half, and the decimal 0.5 would all be equally valid answers for the probability.
A game at a festival challenged people to
throw a baseball through a tire. Of the first 68 participants, three
people won the gold prize, 12 won the silver prize, and 15 won the bronze prize. What is the experimental probability of
not winning any of the three prizes?
Let’s begin this by picking out the key
pieces of information. Three people won the gold prize, 12
people won the silver prize, and 15 people won the bronze prize. However, we’re told that 68 people tried
the game. And if we add together the three, the 12
and the 15 people that won prizes, this will add up to 30, which means that there must be 38
people who didn’t win any prize since 68 subtract 30 gives us 38.
The question, however, is not simply
asking how many people didn’t win a prize, but instead it’s asking for the experimental
probability. We can recall that to calculate the
experimental probability of an event E. This is equal to the number of times E
occurs divided by the total number of trials. We can answer this question using two
different possible methods, but each one will still use the same formula.
In the first method, we can write that
the probability of not winning a prize is equal to the number of non-prize winners divided
by the number of participants. And therefore, as we have 38 people who
didn’t win a prize divided by 68 people in total, this would be the fraction 38 over 68. We can then further simplify this faction
to give 19 over 34. Let’s record this value up here so that
when we clear the screen and try the second method, we can check that both would give the
same result.
For the second method, we’re going to use
the total probability rule, which tells us that the sum of all probabilities is equal to
one. In the method we’ve just seen, we worked
out the probability of not winning a prize. In the second method, we’re going to
calculate the probability of winning a prize and then subtract it from one. So, in our second method, we’re going to
calculate the probability of winning a prize, which is equal to the number of prize winners
divided by the number of participants.
Adding together all the prize winners
give us 30. And since we still have 68 people, then
this will be 30 over 68. We can then simplify this to give us the
fraction 15 over 34. And now, using our total probability
rule, we would have the probability of not winning a prize is equal to one subtract 15 over
34 which is 19 over 34 since we could write one as the fraction 34 over 34. And, therefore, using either method would
give us that the experimental probability of not winning any of the prizes is 19 over
34.
In the next example, we’ll see how we can
apply the experimental probability of a sample to result for the wider population.
A soft drinks factory produces 1400
bottles a day. The factory tested a sample of 400 units
and found that six were defective. By calculating the experimental
probability that a bottle is defective, work out how many defective bottles would be
expected in a day.
So, here, we have a drinks factory, which
produces 1400 bottles a day. They want to check how many are
defective, but they don’t check every single bottle. Instead, they take a sample size of 400
bottles and check those, finding that six of these 400 bottles are defective. We’re asked here to calculate the
experimental probability that a bottle is defective.
We can recall that the experimental
probability of an event E is equal to the number of times E occurs over the total number of
trials. In this question, to work out the
probability of getting a defective bottle, we calculate the number of defective bottles over
the total number of bottles, which will be a six over 400, since we had six defective
bottles in our sample of 400. We now take the probability of getting a
defective bottle in our sample and translate it into the wider population of the 1400
bottles that are made in the factory. And therefore, to work out the number of
defective bottles, we take our probability six out of 400 and multiply it by 1400. We could then simplify this
multiplication into 42 over two, giving us a final answer that 21 bottles would be expected
to be defective each day.
In the next example, we’ll see how a data
set in a table can be used to find experimental probability. We’ll need to be particularly careful in
this question to select the appropriate values.
The table shows the music preferences of
a group of men and women. Calculate the relative frequency of a
randomly selected person being a woman who prefers country music. If necessary, round your answers to three
decimal places. Calculate the relative frequency of a
randomly selected woman preferring rock music. If necessary, round your answers to three
decimal places.
Let’s begin by having a look at the
table. We can see, for example, that there are
13 women who prefer country music and there are 24 women who like rock music. We could, therefore, establish that there
must be 37 women in this group, since that’s the sum of 13 and 24. Equally to calculate the total men in
this group, since there are eight that like country music and 18 that like rock music, then
there must be 26 men in this group. We could also calculate the number of
people who like country music. Since 13 women do and eight men do, then
that’s 21 people in total. Similarly, we can add the column for rock
music to establish that 42 people like rock music.
We can then work out the total number of
people in the group by adding our total men and total women or by adding the totals of those
who like country music and those who like rock music. And so, we can see that either of these
would give us 63 people in the group. So, let’s look at our first question to
calculate the relative frequency.
We can recall that to find the relative
frequency of an event E, this is equal to the number of times E occurs over the total number
of trials. So, to find the relative frequency of
women who like country music, we write the number of women liking country music over the
total number of people. And so, our relative frequency is equal
to 13 over 63. And as we’re asked to write our answers
to three decimal places, we change this fraction into a decimal. And using our calculator, we can evaluate
this as 0.206349 repeating. To round to three decimal places means we
check our fourth decimal digit to see if it is five or more. And as it isn’t, then our answer stays as
0.206. And this is our answer for the first part
of the question.
We can clear some space to answer the
second question. Here, we’re asked to find the relative
frequency of a randomly selected woman preferring rock music. So, to find this relative frequency, we
need the number of women who like rock music or the women rockers. And this time, we write it over the total
number of women and not the total number of people because we’re told that we’re selecting
from the women and not selecting from the group of people. And therefore, our relative frequency as
a fraction will be 24 over 37. As a decimal, this will be equal to 0.648
repeating. As we want to round this to three decimal
places, we can consider that this is equal to 0.648648 and so on. It is that a little bit more easy to
check our fourth decimal digit. And as this digit is five or more, then
our value will round up to 0.649. And so, we have our answer for the second
part of the question.
And now, let’s summarize what we’ve
learned in this video. The experimental probability of an event
E is an estimate of the probability for the event, probability of E, based on data from a
number of trials or experiments. The experimental probability or relative
frequency is given by the relative frequency of event E equals the number of times E occurs
over the total number of trials. We saw in one of our examples that it can
be useful to use the total probability rule, which tells us that the sum of the
probabilities of all possible outcomes is equal to one.
