Experimental probability
Calculate experimental probability from frequency data, predict expected frequencies, and compare experimental and theoretical probability.
Worked examples
Calculating experimental probability
Straightforward
Problem
A spinner is spun 40 times. The results are: red 12 times, blue 18 times, green 10 times. Find the experimental probability of landing on red. Give your answer as a fraction in simplest form.
1
Count the number of times red appeared.
Red appeared 12 times.
2
Write the total number of trials.
Total spins
3
Apply the experimental probability formula and simplify.
Answer
3/10
Predicting expected frequency
Moderate
Problem
In 50 coin flips, heads appeared 30 times. If the coin is flipped 200 times, how many times would you expect to get heads?
1
Find the experimental probability of heads.
2
Multiply the experimental probability by the new number of trials.
Answer
120
Comparing experimental and theoretical probability
Challenging
Problem
A fair six-sided die is rolled 180 times. Lin records the results and the number 2 appears 36 times. (a) Find the experimental probability of rolling a 2. (b) Using theoretical probability, how many times would you expect to roll a 2 in 180 rolls?
1
Calculate the experimental probability from Lin's results.
2
State the theoretical probability of rolling a 2.
A fair die has 6 equally likely outcomes, so .
3
Use theoretical probability to find the expected frequency.
4
Compare the results.
Experimental probability is slightly higher than theoretical probability . This difference is due to chance variation. With more trials, the experimental probability would likely move closer to .
Answer
Experimental probability: 1/5; theoretical expected frequency: 30
Practise
Q1·Straightforward
A coin is flipped 20 times. It lands heads 12 times. What is the experimental probability of getting heads? Give your answer as a fraction in simplest form.
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Explanation
The experimental probability is found by dividing the number of favourable outcomes by the total number of trials:
Q2·Straightforward
A spinner is spun 50 times. It lands on blue 15 times. What is the experimental probability of landing on blue? Give your answer as a fraction in simplest form.
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Explanation
Q3·Straightforward
In 40 draws with replacement from a bag of coloured tiles, green was drawn 10 times. What is the experimental probability of drawing a green tile? Give your answer as a fraction in simplest form.
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Explanation
Q4·Straightforward
A standard die is rolled 30 times. The number 6 appears 5 times. What is the experimental probability of rolling a 6? Give your answer as a fraction in simplest form.
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Explanation
Q5·Moderate
A thumbtack is dropped 100 times and lands point up 60 times. If it is dropped 200 times, how many times would you expect it to land point up?
Explanation
Experimental probability of landing point up:
Expected frequency in 200 drops:
Expected frequency in 200 drops:
Q6·Moderate
A marble is drawn from a bag 40 times (with replacement). A red marble is drawn 8 times. If the marble is drawn 120 times, how many times would you expect to draw a red marble?
Explanation
Experimental probability of red:
Expected frequency in 120 draws:
Expected frequency in 120 draws:
Q7·Moderate
Based on previous spins, the experimental probability of a spinner landing on yellow is . If the spinner is spun 60 times, how many times would you expect it to land on yellow?
Explanation
Expected frequency
Q8·Moderate
In 50 rolls of a die, the number 4 appeared 12 times. If the die is rolled 150 times, how many times would you expect to roll a 4?
Explanation
Experimental probability:
Expected frequency in 150 rolls:
Expected frequency in 150 rolls:
Q9·Challenging
A fair coin is flipped 10 times and lands heads 7 times, giving an experimental probability of . The theoretical probability of heads is . Which statement best explains this difference?
Explanation
A fair coin should land heads with probability , but with only 10 flips, random variation can produce results well above or below this. As the number of trials increases, experimental probability tends to get closer to the theoretical probability.
Q10·Challenging
A fair die is rolled 120 times. Based on theoretical probability, how many times would you expect to roll a number greater than 4?
Explanation
The outcomes greater than 4 are — 2 out of 6 equally likely outcomes:
Expected frequency:
Expected frequency:
Q11·Challenging
A student rolls a die 60 times and the number 2 appears 14 times. What is the experimental probability of rolling a 2? Give your answer as a fraction in simplest form.
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Explanation
The highest common factor of 14 and 60 is 2:
Q12·Challenging
A spinner has 4 equal sections. The theoretical probability of landing on red is . After 20 spins, the experimental probability of red is . If the spinner were spun 1000 times, what would most likely happen to the experimental probability of red?
Explanation
As the number of trials increases, experimental probability tends to approach the theoretical probability. With 1000 spins, the experimental probability of red would most likely be much closer to than it is after only 20 spins.