Experimental probability
Experimental probability, as the name suggests, describes the probability of an event occurring when undertaking experiments or trials. 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
Theoretical probability on the other hand is the 'expected' probability based on knowledge of the system and determining the number of favourable outcomes and number of total possible outcomes mathematically.
$\text{Theoretical Probability}=\frac{\text{Number of favourable outcomes}}{\text{Total number of outcomes}}$Theoretical Probability=Number of favourable outcomesTotal number of outcomes
Worked example
Example 1
a) A coin is tossed 500 times. If 260 of the tosses turn up a tail, what is the experimental probability of getting a tail?
| $Pr\left(\text{Tail}\right)$Pr(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?
| $Pr\left(\text{Tail}\right)$Pr(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 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.
Experimental probabilities can be used as point estimates for the actual probability. For estimates 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.
Experimental probabilities can also be used to test systems with known theoretical probabilities. For example, we could test to see if a coin is biased. For a small number of trials, it would be expected that the probability would vary, but if the coin was tossed many times, the experimental probability should be close to $0.5$0.5. So if a coin is tossed $100$100 times and there is an experimental probability of $0.3$0.3 for obtaining a tail, it could be concluded with reasonable certainty that the coin is biased. How much the experimental probabilities could be expected to vary, and how many trials is sufficient, will be explored 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.
a) What is the experimental probability that a cable at this factory will be faulty?
| $Pr\left(\text{Faulty}\right)$Pr(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 |
b) If $1500$1500 more cables were tested, how many would you expect to be faulty?
Now that we know that approximately $15.2%$15.2% are faulty (from our experimental data), we could expect the same percentage to be faulty from any amount.
So, $15.2%$15.2% of $1500$1500$=$=$0.152\times1500=228$0.152×1500=228
We could expect $228$228 to be faulty from $1500$1500 cables.
Practice questions
QUESTION 1
The following table shows the frequency of the lengths jumped at a long jump competition. The class interval is a range of distances measured in centimeters.
| Class Interval | Frequency |
|---|---|
| $0-39$0−39 | $6$6 |
| $40-79$40−79 | $9$9 |
| $80-119$80−119 | $7$7 |
| $120-159$120−159 | $8$8 |
| $160-199$160−199 | $1$1 |
| $200-239$200−239 | $2$2 |
| Sum | $33$33 |
According to the table, what is the probability that someone jumped more than 119 cm?
What is the probability that someone's jump measured between 80 and 159cm (inclusive)?
QUESTION 2
The table attached presents the results of multiple coin tosses:
| Coin Toss | Frequency |
|---|---|
| Heads | $52$52 |
| Tails | $48$48 |
How many times was the coin tossed?
What is the relative frequency of tossing a head?
- What is the experimental probability of tossing a tail?
If the coin was tossed 600 times, how many times would you expect it to land on a head?
If the coin was tossed 800 times, how many times would you expect it to land on a tail?
QUESTION 3
$1000$1000 transistors were tested at a factory, and $12$12 were found to be faulty.
What is the experimental probability that a transistor at this factory will be faulty?
If another $5000$5000 transistors were tested, how many of these would you expect to be faulty?