In order to predict the future, we sometimes need to determine the probability by running experiments, or looking at data that has already been collected. This is called experimental probability, since we determine the probability of each outcome by looking at past events.
Experimental probability, as the name suggests, describes the probability of an event occurring when undertaking experiments or trials. Another name for experimental probability is relative frequency. It can be calculated as follows:
$\text{Experimental Probability}=\frac{\text{Number of favourable outcomes}}{\text{Total number of trials}}$Experimental Probability=Number of favourable outcomesTotal number of trials
Worked example
Example 1
a) A coin is tossed $500$500 times. If $260$260 of the tosses turn up a tail, what is the experimental probability of getting a tail?
| $P\left(\text{Tail}\right)$P(Tail) | $=$= | $\frac{\text{Number of favourable outcomes}}{\text{Total number of trials}}$Number of favourable outcomesTotal number of trials |
| $=$= | $\frac{260}{500}$260500 | |
| $=$= | $0.52$0.52 |
b) What is the theoretical probability of obtaining a tail when tossing a coin?
| $P\left(\text{Tail}\right)$P(Tail) | $=$= | $\frac{\text{Number of favourable outcomes}}{\text{Total number of outcomes}}$Number of favourable outcomesTotal number of outcomes |
| $=$= | $\frac{1}{2}$12 | |
| $=$= | $0.5$0.5 |
As the number of trials increases, the experimental probability will tend towards the theoretical probability. The graph below shows the results of tossing a coin $50$50 times. As we can see the probability varies significantly at the start but as we increase the number of coin tosses the probability settles and approaches the theoretical probability of $0.5$0.5.
Experimental probability is very important in cases where the theoretical probability cannot be calculated. Such as research or experiments in environmental sciences, behavioural sciences, economics, medicine and marketing. For example: if we want to know the probability of catching a fish over a certain size from a large lake, how could we find the probability? We couldn't know the theoretical probability unless we caught and measured every fish in the lake. However, working from previous data of sizes of fish caught in the lake we can use the experimental probability for an estimate of the actual probability.
For our estimate to be reasonable the number of trials must be sufficiently large. There are many other experimental design factors that come into play to ensure the sample is representative of the population.
We can also use experimental probabilities to test systems that we know the theoretical probability of. For example, we could test to see if a coin is biased. For a small number of trials we would expect the probability to vary, but if we tossed the coin many times we would expect the experimental probability to be close to $0.5$0.5. So if we tossed a coin $100$100 times and had an experimental probability of $0.3$0.3 for obtaining a tail, we could conclude with reasonable certainty that the coin is biased. How much you expect the experimental probabilities to vary and how many trials is sufficient can be looked at in depth in further studies of statistics.
Worked example
example 2
$500$500 cables were tested at a factory, and $76$76 were found to be faulty. What is the experimental probability that a cable at this factory will be faulty?
| $P\left(\text{Faulty}\right)$P(Faulty) | $=$= | $\frac{\text{Number of favourable outcomes}}{\text{Total number of trials}}$Number of favourable outcomesTotal number of trials |
| $=$= | $\frac{76}{500}$76500 | |
| $=$= | $0.152$0.152 |
We can simplify this fraction to $\frac{19}{125}$19125, or convert it to a percentage which is $15.2%$15.2%.
Reflect: Here we cannot know the theoretical probability that a cable is faulty but the experimental probability can be used to estimate the chance of a cable from this factory being faulty.
Use the following applet to explore relative frequency by simulating the result of rolling a six-sided die.
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Practice questions
Question 1
After flipping a fair coin $70$70 times, the relative frequency of heads is found to be $\frac{52}{70}$5270.
What is the probability of the next flip being a tail?
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If the experiment were to be repeated an infinite number of times, what would you expect the relative frequency of heads to be?
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Question 2
A die is rolled $60$60 times and the results are recorded in the following table:
| Number | Frequency |
|---|---|
| $1$1 | $10$10 |
| $2$2 | $12$12 |
| $3$3 | $8$8 |
| $4$4 | $10$10 |
| $5$5 | $8$8 |
| $6$6 | $12$12 |
What is the experimental probability of rolling a $6$6 with this die?
Express your answer in simplest form.
What is the experimental probability of rolling a $3$3 or higher with this die?
Express your answer in simplest form.
What is the experimental probability of rolling a $3$3 or lower with this die? Express your answer in simplified form.
Question 3
Consider the histogram showing outcomes of a coin toss experiment.
How many times was the coin flipped?
What was the relative frequency of heads in this experiment?