Experimental Probability
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- Experimental Probability - Video - Experimental Probability Watch
Theory
Experimental probability is found by doing an experiment many times and counting how often an event happens: \(P(E) = \dfrac{\text{frequency of } E}{\text{number of trials}}\). This Year 7 maths page shows how to read frequency tables, compare results with the theoretical probability, predict expected numbers and spot bias.
A trial is one go of an experiment: one toss, one roll or one spin. The frequency of an outcome is the number of times it happened.
The experimental probability (also called the observed probability or relative frequency) is \(P(E) = \dfrac{\text{frequency of } E}{\text{total number of trials}}\). In a table, the total number of trials is the total frequency (add all the frequencies).
The theoretical probability is the chance you expect under fair conditions: a fair coin gives \(P(\text{head}) = \dfrac{1}{2}\), and a fair die gives \(P(\text{any one number}) = \dfrac{1}{6} = 0.1\dot{6}\).
If the division does not end, round it: \(7\) heads in \(30\) tosses gives \(7 \div 30 = 0.2333\ldots \approx 0.23\) (\(2\) decimal places), or about \(23\%\).
The expected number is a prediction: expected \(=\) probability \(\times\) number of trials. A sample’s relative frequency can predict a larger group.
The more trials you do, the closer the relative frequency usually gets to the theoretical probability. A random number generator can run hundreds of trials quickly. If the results stay far from the theory after many trials, the object may be biased (unfair).
Two rules:
How to find an experimental probability
- Add all the frequencies to get the number of trials.
- Add the frequencies of the favourable outcomes.
- Divide: frequency over trials. Simplify, or round as asked.
How to predict
- Choose the probability (theoretical, or experimental from a sample).
- Multiply by the number of trials (or the size of the group).
| \(\dfrac{15}{60}\) | \(=\) | \(\dfrac{1}{4}\) |
| \(\) | \(=\) | \(0.25\) |
\(\dfrac{1}{4}\), or \(0.25\)
Outcome: \(1, 2, 3, 4, 5, 6\)
Frequency: \(12, 13, 13, 15, 11, 16\)
Find the relative frequency, as a decimal, of
(i) a \(6\)
(ii) an even number
| \(12 + 13 + 13\) | \(=\) | \(38\) |
| \(38 + 15 + 11 + 16\) | \(=\) | \(80\) |
| \(\dfrac{16}{80}\) | \(=\) | \(0.2\) |
| \(13 + 15 + 16\) | \(=\) | \(44\) |
| \(\dfrac{44}{80}\) | \(=\) | \(0.55\) |
(i) \(0.2\)
(ii) \(0.55\), close to the theoretical \(0.5\)
Sum: \(2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\)
Frequency: \(2, 3, 5, 6, 8, 10, 9, 7, 5, 3, 2\)
Find the experimental probability of
(i) a sum of \(10\)
(ii) an odd sum
(iii) a sum less than \(6\)
| \(2 + 3 + 5 + 6 + 8 + 10\) | \(=\) | \(34\) |
| \(34 + 9 + 7 + 5 + 3 + 2\) | \(=\) | \(60\) |
| \(3 + 6 + 10 + 7 + 3\) | \(=\) | \(29\) |
| \(2 + 3 + 5 + 6\) | \(=\) | \(16\) |
(i) \(\dfrac{5}{60} = \dfrac{1}{12}\)
(ii) odd sums are \(3, 5, 7, 9, 11\): \(\dfrac{29}{60}\)
(iii) sums \(2, 3, 4, 5\): \(\dfrac{16}{60} = \dfrac{4}{15}\)
(i) the theoretical probability of blue
(ii) the expected number of blues
(iii) the experimental probability of blue
| \(\dfrac{3}{5} \times 50\) | \(=\) | \(30\) |
| \(\dfrac{34}{50}\) | \(=\) | \(\dfrac{17}{25}\) |
(i) \(\dfrac{3}{5}\)
(ii) \(30\)
(iii) \(\dfrac{17}{25}\). A little more than expected; normal for \(50\) spins.
(i) How many times did green come up?
(ii) Estimate the number of red counters.
| \(0.45 \times 60\) | \(=\) | \(27\) |
| \(60 - 27 - 15\) | \(=\) | \(18\) |
| \(0.45 \times 20\) | \(=\) | \(9\) |
(i) \(18\)
(ii) about \(9\) red counters
(ii) Leo rolls a die \(300\) times and gets \(95\) twos. Is the die fair?
| \(\dfrac{4}{250} \times 8000\) | \(=\) | \(128\) |
| \(\dfrac{1}{6} \times 300\) | \(=\) | \(50\) |
(i) about \(128\) faulty batteries
(ii) A fair die would give about \(50\) twos. \(95\) is far more, over many rolls (\(\dfrac{95}{300} \approx 0.32\), not about \(0.17\)), so the die is probably biased.
Common pitfalls
Frequently asked questions
What is experimental probability?
The number of times an event happened divided by the number of trials.
What is relative frequency?
Another name for experimental probability: frequency divided by the total frequency.
How do you find an expected number?
Multiply the probability by the number of trials. A fair coin tossed \(80\) times: about \(40\) heads.
Why is experimental probability different from theoretical probability?
Chance results vary. With more trials they usually get closer.
How can you tell if a die is biased?
After many rolls, one number comes up far more or far less often than expected.