Experimental probability
Experimental probability, as the name suggests, describes the probability when undertaking experiments or trials. Another name for experimental probability is relative frequency.
We can calculate probabilities by constructing a fraction like this: $\frac{\text{what you want}}{\text{total }}$what you wanttotal which written more formally as
$P\left(\text{event}\right)=\frac{\text{frequency of event}}{\text{total number trials}}$P(event)=frequency of eventtotal number trials
Law of large numbers
The law of large numbers states that 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 1
$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?
Solution: $\frac{\text{frequency of the event }}{\text{total number of trials }}=\frac{76}{500}$frequency of the event total number of trials =76500
We can simplify this fraction to $\frac{19}{125}$19125, or convert it to a percentage which is $76\div500\times100=15.2%$76÷500×100=15.2%
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
A coin was flipped $178$178 times with $93$93 tails recorded.
What is the exact experimental probability of flipping tails with this coin?
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?