In this explainer, we will learn how to interpret a data set by finding and evaluating the
experimental probability.
Probability refers to the likelihood or chance of an event occurring. In experimental
probability, we make estimates for the likelihood or chance of something occurring based on
the results of a number of experiments or trials.
Let us look at an example to illustrate what we mean by this. We will calculate the
experimental probability using data in a table.
Example 1: Experimental Probability from a Table
The table shows the results of a survey that asked 20 students about their favorite
breakfast.
What is the probability that a randomly selected student prefers eggs?
Answer
As we are using data collected in a survey, this is classified as experimental data. The
20 students asked about their favorite breakfast are the 20 trials in our experiment.
Since 10 out of the 20 students prefer eggs, the probability that a randomly selected
student prefers eggs is
The probability that a student selected at random prefers eggs for breakfast is therefore
0.5. Or converting this to a percentage, we can say there is a 50% chance that a student
selected at random prefers eggs for breakfast. (Multiplying the probability 0.5 by 100
gives us the 50%.)
We will look at some more examples, but before we do this let us list the main points of
interest for experimental probability.
Experimental Probability: Main Points
The experimental probability of event is an estimate of the probability
for the event , based on data from a number of trials or
experiments. So, for example, if we use data collected in a survey to estimate a
probability, this would be classed as experimental probability.
The experimental probability of an event is often also called the relative frequency
of the event and is given by
As with any probability, if we have the experimental probability of an event
, we can find the probability that does
not occur, , by using
the total probability rule: So
.
The next example demonstrates how to calculate the relative frequency of an event.
Example 2: Experimental Probability and Relative Frequency
A coin was tossed 200 times and the number of tails observed was 102. Calculate the
relative frequency of getting a heads. Calculate the answer to three decimal
places.
Answer
As the coin was tossed 200 times, this means there were 200 โtrials.โ We
are looking for the relative frequency of getting a heads, but we know that 102 of the
throws resulted in tails. So to begin, we will calculate the relative frequency of getting
a tails, which is
Hence, the relative frequency of getting a tails is . We
can use this to find the relative frequency of getting a heads, which can be worked out in
two different ways with the information we have.
Method 1 uses the rule that the sum of the probabilities for all possible
outcomes is equal to 1. We have just worked out the relative frequency (or
probability) of getting a tails. This is . Since there
is only one other possible outcome (heads), subtracting the โtailsโ
probability from 1 gives us the probability (or relative frequency) of getting a head:
To three decimal places, .
Method 2 uses the number of tails that occurred and the total number of
trials to calculate directly the number of heads out of all the trials. We then use
this to calculate the relative frequency or probability of a heads:
As there
were 98 heads out of 200 trials, the relative frequency of getting a heads is
The relative frequency of getting a heads
is therefore 0.490 to 3 decimal places.
In our next example, we calculate the experimental probability for an event.
Example 3: Experimental Probability
A game at a festival challenged people to throw a baseball through a tire. Of the
first 68 participants, 3 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?
Answer
In order to find the experimental probability of not winning any of the three
prizes, let us first summarize the information that we have:
The total number of trials is the total number of participants, which is 68. We know also
that, of these 68 participants, 30 won a prize. If 30 won a prize, then participants did not win a prize and we can use this information to
calculate the experimental probability of not winning any of the three prizes:
Hence, the experimental probability of not winning a prize is
.
Note that we could also have used the total probability rule to answer this question
(i.e., that the sum of the probabilities for all possible outcomes is equal to 1). With 68
participants, 30 of whom won a prize, the probability of winning a prize is
. By the total probability rule, the
probability of not winning a prize is . So
Our next example demonstrates the use of tabular information in finding relative
frequencies.
Example 4: Using Information from a Table to Find Relative Frequencies
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 answer to 3 decimal places.
Calculate the relative frequency of a randomly selected woman preferring rock
music. If necessary, round your answer to 3 decimal places.
Answer
In order to find the relative frequencies, our first step is to calculate the total
number of trials. We can work out the totals for each category as in the table below.
Part 1
To calculate the relative frequency of a randomly selected person being a woman who
prefers country music, we must find the number of people falling into this category and
also the total number of people. (The total number of people is the number of trials.)
We can see that the number of women who prefer country music is 13 and that the total
number of people whose preferences were recorded (i.e., the number of trials) is 63. The
relative frequency (or probability ) of a randomly selected person
being a woman (W) who prefers country music (C) is, therefore,
The relative frequency, to three decimal places, of randomly selecting a woman who
prefers country music is therefore 0.206. Or as a percentage, there is a 20.6% () chance that a person selected at random is a woman and
prefers country music.
Part 2
To calculate the relative frequency of a randomly selected woman preferring rock music,
we need to know how many women there were in total and the number of those who prefer rock
music.
There was a total of 37 women, 24 of whom prefer rock music. So the relative frequency of
women who prefer rock music is
Hence, the relative frequency of a randomly selected woman preferring rock music is 0.649
to three decimal places. Or, as a percentage, there is approximately a 65%
() chance that a woman selected at random
prefers rock music.
Note that there is a subtle difference between the wording of the two parts of this
question. Part 1 asks for the relative frequency of a โrandomly selected person
being a woman who prefers country music.โ And part 2 refers to a โrandomly
selected woman preferring rock music.โ
The distinction is that, in part 2, we are making a random selection only from the
women, whereas in part 1 we are randomly selecting from all of the people whose
preferences were recorded. That is why in the solution to part 2 our denominator is the
total number of women only, and in part 1 the denominator is the overall total, that is,
both men and women.
Sometimes when looking at experimental probability, we may not be able to directly
perform the experiment we would like but it is possible to model the situation. In the
next example, we will consider how an experiment might be constructed to calculate
experimental probability.
Example 5: Constructing an Experiment for Experimental Probability
One out of every six students in a seventh-grade class is left handed. Which of the
following could be used to find the experimental probability that we will get a
left-handed student when choosing randomly?
Using a spinner with four colors: red represents right handed, and yellow, blue,
and green represent left handed.
Using a coin: heads is right handed, and tails is left handed.
Using a number cube: even numbers represent right handed and odd numbers
represent left handed.
Using a number cube: landing on 1 represents left handed, and landing on
2โ6 represents right handed.
Answer
Our information is that one in every six students is left handed. This means that for
every individual left-handed student we expect there to be 5 right-handed students. So we
are looking for an experiment with this ratio in its design. Let us consider each of the
options separately and see if they fit the bill.
Using a spinner with four colors: red represents right handed, and yellow, blue, and
green represent left handed.
This option will
not fit. The spinner has only four colors and the ratio specified in this option is 1
right handed to 3 left handed, so there is no possibility that we can fit the ratio 1
left handed to 5 right handed to this scenario.
Using a coin: heads is right handed, and tails is left handed.
This option does not fit the bill either. The
coin has only two possible outcomes, in the ratio 1 to 1. So we cannot fit our ratio 1
to 5 in this scenario.
Using a number cube: even numbers represent right handed and odd numbers represent
left handed.
Using a number cube looks
promising as it has 6 faces; however, the specification that โeven numbers
represent right-handed and odd numbers represent left-handed studentsโ does not
work for us. As there are equal numbers of odd and even faces, this has reduced our
ratio again to 1 to 1. For every even face, there is an odd face.
Using a number cube: landing on 1 represents left handed, and landing on 2โ6
represents right handed.
In this case, we have
a number cube where 1 represents left handed and the numbers 2โ6 represent
right handed. This works for us since the ratio of possibilities is 1 to 5. From 2 to
6 (including 2 and 6), there are 5 numbers. So this scenario could be used to find the
experimental probability that we will get a left-handed student when choosing
randomly.
Let us remind ourselves of the key ideas related to experimental probability
Key Points
The experimental probability of event is an estimate of the
probability for the event , based on data from a
number of trials or experiments.
The experimental probability of an event is also called the relative frequency of the
event and is given by
When calculating probabilities, we often use the โtotal probability ruleโ:
Join Nagwa Classes
Attend live sessions on Nagwa Classes to boost your learning with guidance and advice from an expert teacher!