Video Transcript
In this video, we will learn how to
interpret a dataset by finding and evaluating experimental probability.
When we are calculating
probability, we’re really determining the likelihood that this event will occur. There are two main ways in which we
can estimate the probability of an event. So let’s consider these first. One way in which we can estimate
the probability is by considering the attributes or physical properties of the event
in question. For example, if we wanted to
calculate the probability of rolling a five on a fair die, we would consider the
number of sides on the die. The probability of one-sixth would
be the theoretical probability of rolling a five.
But sometimes we can’t use
theoretical probability, for example, if we knew that this die was unfair. This is an occasion where we could
use experimental probability. This is when we perform an
experiment, for example, throwing a die. We say that we perform repeated
trials and record the outcome, that’s the result, of each trial. We can then calculate experimental
probability using the formula that this is equal to the number of trials in which
the outcome occurs over the total number of trials. Let’s now see how we can apply this
in a practical example.
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?
We can begin this question by
having a look at the table. We can see that out of 20 students,
10 students said that eggs were their favorite breakfast, two students said that
cereal was, and eight students said that toast was their favorite. When it comes to calculating the
probability that a student prefers eggs, we’re really considering this as
experimental probability. Although it doesn’t appear to be a
traditional experiment, there has been a repeated trial of asking different students
for their favorite breakfast and then a recording of the outcomes. That’s the answers that the
students gave. The standard formula that we might
use to calculate experimental probability can be given that this is equal to the
number of trials in which the outcome occurs over the total number of trials.
However, we can rewrite this
according to the context. We want to calculate the
probability that a student prefers eggs. So that can be calculated as the
number of students who preferred eggs over the total number of students
surveyed. Then, from the table, we can see
that there were a total of 10 students who preferred eggs. And we are told that 20 students
were surveyed in total. Notice that even if we weren’t
given the figure of 20, we could calculate this from the sum of the frequencies,
that’s the number of students, of 10 plus two plus eight. So we now have that the probability
of a student preferring eggs is 10 over 20. But of course we know that this
fraction can be simplified further to one-half. And so we can give the answer that
the probability that a randomly selected student prefers eggs is one-half.
In the next example, we’ll see how
to calculate the probability when the outcomes of an experiment are given as a
list.
Isabella creates a three-sided
spinner using the colors red, green, and blue. She spins the spinner and records
the following results: red, blue, red, green, green, green, red, red, red,
green. Calculate the experimental
probability of spinning green on this spinner.
We are told that Isabella has a
three-sided spinner with three colors: red, green, and blue. The outcomes of the experiment to
spin the spinner have been recorded. We then need to calculate the
experimental probability of spinning green. The experimental probability of an
event is the ratio of the number of outcomes in which this event occurs to the total
number of trials in the experiment. So here we’re calculating the
experimental probability of spinning green. This is equal to the number of
times green was spun over the total number of spins.
Using the set of results, we can
see that green was spun four times. And by counting the number of
results, we can determine that the number of spins must be 10. The fraction four-tenths can be
fully simplified to two-fifths. We can therefore give the answer
that the experimental probability of spinning green is two-fifths. However, as probabilities can be
given as fractions, decimals, or percentages, then 0.4 or 40 percent would also be
valid answers.
We have now seen how we can
determine experimental probability when the data is presented as a table and as a
set of results. In the next example, we’ll see how
we can use a bar graph to calculate experimental probability.
The graph shows the results of an
experiment in which a die was rolled 26 times. Find the experimental probability
of rolling a two. Give your answer as a fraction in
its simplest form.
Here, we can see that we have a bar
graph representing the number of times the values one to six were rolled on a
die. For example, number one was rolled
four times. Number two was rolled eight
times. We can calculate the experimental
probability of rolling a two as the number of times two was rolled over the total
number of rolls. If we look at the bar chart, even
if we weren’t given the figure of eight, we could read that the top of bar two comes
to the value of eight. In other words, the number two was
rolled eight times. We are then given that the die was
rolled 26 times. However, even if we weren’t given
this total number of rolls, we could calculate it by adding four, eight, eight,
three, one, and two. This would give us a value of
26. The probability of rolling a two is
eight over 26. But as these are both even numbers,
we know that this fraction will simplify further to four over 13. And that’s our fraction in its
simplest form for the experimental probability of rolling a two.
Let’s now see another example.
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?
Although it may at first appear
that there are just three different outcomes in this game, there are in fact
four. It’s possible to win gold, silver,
or bronze prizes or no prizes. Given the information that three
people won gold, 12 won silver, and 15 won the bronze prize out of a total of 68
participants, we can calculate the number of participants who won no prizes. It will be equal to 68 minus three
plus 12 plus 15. That’s 38 people. We can then use this value to find
the experimental probability of not winning any prizes, since this is equal to the
number of participants who won no prizes over the total number of participants. 38 people won no prizes, and there
were 68 participants. This simplifies to give us the
answer that the experimental probability of not winning any of the three prizes is
19 over 34.
In the final question, we will use
a given experimental probability and a value for the number of outcomes to calculate
the total number of trials in an experiment.
The experimental probability that a
coin lands on tails is three-sevenths. If the coin landed on tails 30
times, how many times was it tossed in the experiment?
In this question, we are given the
experimental probability that a coin lands on tails. This probability has been
calculated using the data from an experiment with repeated trials of tossing a
coin. We are also given that the number
of outcomes of landing on tails, that’s the number of times the coin landed on
tails, is 30. We can recall that, in general,
experimental probability is calculated as the number of trials in which the outcome
occurs over the total number of trials. In the context of this problem, we
would say that the experimental probability of a coin landing on tails is equal to
the number of times the coin landed on tails over the total number of times the coin
was tossed.
Usually, when we apply this
formula, we’re doing it to calculate the experimental probability when we are given
the two values on the right-hand side. However, here we know the
experimental probability, we know the number of times the coin landed on tails, and
we want to calculate the total number of times the coin was tossed. We can then substitute in the
values of three-sevenths for the experimental probability and 30 for the number of
times the coin landed on tails. We can then solve this by cross
multiplying. Three times the total number of
times the coin was tossed is equal to 30 times seven, and 30 times seven is equal to
210. We can then divide both sides by
three. 210 divided by three is equal to
70. The answer is that the coin was
tossed 70 times in this experiment.
We can now summarize the key points
of this video. Firstly, we saw that experiments
can be used to estimate the probability of an event occurring. To gather data to calculate
experimental probability, we perform repeated trials and record the outcome of each
trial. It’s worth noting that the more
trials we perform, the more accurate an estimate of results we will get. The experimental probability of an
event is the ratio of the number of outcomes in which a specified event occurs to
the total number of trials in an experiment. We can remember this in the general
formula that experimental probability of an event is equal to the number of trials
in which the outcome occurs over the total number of trials. And finally, as we saw across the
range of our examples, we can calculate experimental probability from data presented
in the form of a statement, a set of results, a table, or a graph.
