Experimental Probability
Experimental probability (also called empirical probability) is the probability of an event calculated from the results of an actual experiment or trial. Unlike theoretical probability (which is based on reasoning), experimental probability is based on what actually happened.
For example, if you toss a coin 50 times and get heads 23 times, the experimental probability of heads = 23/50 = 0.46. This may differ from the theoretical probability of 1/2 = 0.5, but as the number of trials increases, experimental probability approaches the theoretical value.
In real life, many situations do not have equally likely outcomes. A biased coin, a loaded die, the chance of rain tomorrow, or the probability of a cricket batsman hitting a six — these cannot be calculated using theoretical probability alone. For such situations, we collect data from actual experiments or observations and calculate the experimental probability.
Experimental probability forms the basis of statistics — the branch of mathematics that deals with collecting, organising, and interpreting data. Weather forecasts, medical research, quality control in factories, and sports analytics all rely on experimental probability.
In this topic, you will learn the formula for experimental probability, how to calculate it from frequency tables, the relationship between experimental and theoretical probability, the Law of Large Numbers, and how to interpret probability values in real-world contexts.
What is Experimental Probability?
Definition: The experimental probability of an event E is defined as:
P(E) = Number of times event E occurs / Total number of trials
Where:
- P(E) = experimental probability of event E
- Number of times E occurs = frequency of the event
- Total number of trials = total number of times the experiment was performed
Key facts:
- Experimental probability is always between 0 and 1 (inclusive).
- P(E) = 0 means the event never occurred in the experiment.
- P(E) = 1 means the event occurred every single time.
- The sum of probabilities of all possible outcomes = 1.
Difference from theoretical probability:
- Theoretical probability = Number of favourable outcomes / Total possible outcomes (based on reasoning).
- Experimental probability = Number of times event happened / Total trials (based on actual data).
- As the number of trials increases, experimental probability gets closer to theoretical probability.
Methods
Steps to find experimental probability:
- Perform the experiment (or collect data from an existing experiment).
- Record the outcomes and count the frequency of each outcome.
- Count the total number of trials.
- Apply the formula: P(E) = Frequency of event / Total trials.
- Express the answer as a fraction, decimal, or percentage.
Recording data:
Use a frequency table:
- Column 1: Outcome
- Column 2: Tally marks
- Column 3: Frequency (count)
Converting between forms:
- Fraction to decimal: divide numerator by denominator.
- Fraction to percentage: multiply by 100%.
- Example: P(E) = 3/10 = 0.3 = 30%.
Important principle — Law of Large Numbers:
- As the number of trials increases, experimental probability approaches theoretical probability.
- 10 coin tosses may give 70% heads. But 10,000 coin tosses will give close to 50% heads.
- More trials = more reliable experimental probability.
Solved Examples
Example 1: Example 1: Coin toss
Problem: A coin is tossed 80 times. Heads appears 36 times. Find the experimental probability of getting (a) heads and (b) tails.
Solution:
Given:
- Total tosses = 80
- Heads = 36
- Tails = 80 − 36 = 44
(a) P(Heads):
- = 36/80 = 9/20 = 0.45
(b) P(Tails):
- = 44/80 = 11/20 = 0.55
Verification: 9/20 + 11/20 = 20/20 = 1 ✓
Answer: P(Heads) = 9/20, P(Tails) = 11/20.
Example 2: Example 2: Dice rolling
Problem: A die is rolled 60 times with these results: 1 appears 8 times, 2 appears 12 times, 3 appears 10 times, 4 appears 9 times, 5 appears 11 times, 6 appears 10 times. Find the probability of getting (a) 2, (b) an even number.
Solution:
(a) P(2):
- = 12/60 = 1/5
(b) P(even number) — even numbers are 2, 4, 6:
- Frequency = 12 + 9 + 10 = 31
- P(even) = 31/60
Answer: P(2) = 1/5, P(even) = 31/60.
Example 3: Example 3: Survey data
Problem: In a survey of 200 students, 70 preferred Cricket, 50 preferred Football, 40 preferred Badminton, and 40 preferred other sports. Find the probability that a randomly selected student prefers Football.
Solution:
Given:
- Total students = 200
- Football = 50
P(Football):
- = 50/200 = 1/4 = 0.25 = 25%
Answer: P(Football) = 1/4 or 25%.
Example 4: Example 4: Defective items
Problem: A factory produces 500 bulbs. On testing, 25 are found defective. Find the probability that a randomly chosen bulb is (a) defective, (b) non-defective.
Solution:
Given:
- Total bulbs = 500, Defective = 25
- Non-defective = 500 − 25 = 475
(a) P(defective):
- = 25/500 = 1/20 = 0.05 = 5%
(b) P(non-defective):
- = 475/500 = 19/20 = 0.95 = 95%
Answer: P(defective) = 1/20, P(non-defective) = 19/20.
Example 5: Example 5: Drawing from a bag
Problem: A bag has red, blue, and green balls. In 50 random draws (with replacement): red comes 18 times, blue comes 20 times, green comes 12 times. Find the experimental probability of drawing each colour.
Solution:
Total draws: 50
Probabilities:
- P(Red) = 18/50 = 9/25
- P(Blue) = 20/50 = 2/5
- P(Green) = 12/50 = 6/25
Verification: 9/25 + 10/25 + 6/25 = 25/25 = 1 ✓
Answer: P(Red) = 9/25, P(Blue) = 2/5, P(Green) = 6/25.
Example 6: Example 6: Weather data
Problem: In the last 90 days, it rained on 18 days. Find the experimental probability that it will rain on any given day.
Solution:
Given:
- Total days = 90, Rainy days = 18
P(Rain):
- = 18/90 = 1/5 = 0.2 = 20%
Answer: P(Rain) = 1/5 or 20%.
Example 7: Example 7: Blood group data
Problem: In a sample of 150 people: Blood group A = 45, B = 36, AB = 15, O = 54. Find the probability of a person having blood group B.
Solution:
Given:
- Total = 150, Blood group B = 36
P(B):
- = 36/150 = 6/25 = 0.24
Answer: P(Blood group B) = 6/25.
Example 8: Example 8: Spinner problem
Problem: A spinner with 4 colours is spun 100 times. Results: Red 28, Blue 32, Green 22, Yellow 18. Find (a) P(Blue) and (b) P(not Red).
Solution:
(a) P(Blue):
- = 32/100 = 8/25
(b) P(not Red):
- = 1 − P(Red) = 1 − 28/100
- = 72/100 = 18/25
Answer: P(Blue) = 8/25, P(not Red) = 18/25.
Example 9: Example 9: Cricket batting data
Problem: A batsman faces 120 balls in a match. He hits a boundary on 15 balls and scores runs on 45 balls. Find (a) P(boundary) and (b) P(no run).
Solution:
(a) P(boundary):
- = 15/120 = 1/8
(b) P(no run):
- Balls with runs = 45 (including the 15 boundaries)
- Balls with no run = 120 − 45 = 75
- P(no run) = 75/120 = 5/8
Answer: P(boundary) = 1/8, P(no run) = 5/8.
Example 10: Example 10: Comparing experimental and theoretical
Problem: A coin is tossed 200 times. Heads appears 112 times. Compare the experimental probability with the theoretical probability.
Solution:
Experimental probability:
- P(Heads) = 112/200 = 0.56 = 56%
Theoretical probability:
- P(Heads) = 1/2 = 0.50 = 50%
Comparison:
- Experimental (56%) is close to but not exactly equal to theoretical (50%).
- Difference = 6 percentage points.
- With more trials (say 10,000), the experimental value would get even closer to 50%.
Answer: Experimental P(Heads) = 0.56, Theoretical P(Heads) = 0.50. They are close but not equal due to randomness in a limited number of trials.
Real-World Applications
Real-world applications of experimental probability:
- Weather forecasting: Meteorologists use historical data to predict rain probability. "70% chance of rain" is based on experimental probability from past weather patterns.
- Medicine: Drug effectiveness is tested through clinical trials. If 800 out of 1000 patients recover, the probability of recovery = 80%.
- Quality control: Factories test samples to estimate defect rates. If 3 out of 500 products fail, defect probability = 0.6%.
- Insurance: Insurance companies use historical accident data to calculate premiums — this is experimental probability applied to risk assessment.
- Sports: A batsman's batting average is experimental probability. Scoring 50 runs in 100 balls gives a strike rate of 50%.
- Elections: Exit polls use experimental probability — sampling voters to predict election outcomes.
- Genetics: Mendel's experiments on pea plants used experimental probability to discover the laws of inheritance.
Key Points to Remember
- Experimental probability = Number of times event occurs / Total number of trials.
- It is based on actual data from experiments, not theoretical reasoning.
- P(E) always lies between 0 and 1 (inclusive).
- P(not E) = 1 − P(E).
- Sum of probabilities of all outcomes = 1.
- Experimental probability varies between different experiments.
- As the number of trials increases, experimental probability approaches theoretical probability (Law of Large Numbers).
- Experimental probability is useful when theoretical probability is difficult to calculate.
- Results should be recorded in a frequency table.
- Probability can be expressed as a fraction, decimal, or percentage.
Practice Problems
- A coin is tossed 150 times and heads appears 82 times. Find the experimental probability of heads and tails.
- A die is thrown 100 times. The number 3 appears 19 times. Find P(3) and P(not 3).
- In a survey of 400 families, 120 have exactly 2 children. Find the probability that a randomly chosen family has 2 children.
- A factory tests 1000 items and finds 35 defective. Find the probability that a random item is defective.
- A spinner is spun 80 times: Red 22, Blue 28, Green 30. Find the probability of landing on each colour.
- In 365 days, a city had 45 rainy days. Find the probability that a randomly chosen day is (a) rainy, (b) not rainy.
- A bag contains different coloured balls. In 60 draws (with replacement): White 15, Black 25, Red 20. Find P(Black).
- Two coins are tossed 100 times. Results: 2 Heads: 24, 1 Head: 52, 0 Heads: 24. Find the probability of getting exactly 1 Head.
Frequently Asked Questions
Q1. What is experimental probability?
Experimental probability is the probability of an event calculated from actual experiment results. P(E) = Number of times E occurs / Total number of trials.
Q2. How is experimental probability different from theoretical probability?
Theoretical probability uses reasoning (favourable outcomes / total outcomes). Experimental probability uses actual data (frequency / total trials). Theoretical is calculated; experimental is observed.
Q3. Can experimental probability be greater than 1?
No. Since the number of times an event occurs cannot exceed the total trials, experimental probability is always between 0 and 1 (inclusive).
Q4. Why does experimental probability change between experiments?
Because random events produce different outcomes each time. Tossing a coin 20 times might give 12 heads one time and 9 heads another time. Only with very large trials do results stabilise.
Q5. What is the Law of Large Numbers?
As the number of trials increases, the experimental probability approaches the theoretical probability. 10 coin tosses may give 70% heads, but 10,000 tosses will give close to 50%.
Q6. When is experimental probability more useful than theoretical?
When outcomes are not equally likely (biased coin), when the experiment is too complex for theory, or when working with real-world data (weather, medicine, surveys).
Q7. What is P(not E)?
P(not E) = 1 − P(E). If the probability of rain is 0.3, then the probability of no rain is 1 − 0.3 = 0.7.
Q8. Does experimental probability predict the future?
Not exactly. It gives an estimate based on past data. The more data you have, the better the estimate. But individual future events remain uncertain.
Q9. What is 'with replacement' in probability experiments?
After drawing an item from a bag, you put it back before the next draw. This keeps the total number of items constant, so each draw is independent.
Q10. How many trials are needed for reliable results?
There is no fixed number, but more trials give more reliable results. Generally, at least 30-50 trials are recommended for basic experiments. For precise results, hundreds or thousands of trials are used.
