Empirical
Probability

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Concept
Definition
Formula, key terms, comparison with classical probability, and worked examples.
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Practice
Solve Step-by-Step
8 guided problems โ€” answer each step to build the full empirical probability solution.
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Empirical Probability Formula
P(E) = Number of times event E occurred Total number of trials (experiments)
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What is Empirical Probability?
Empirical Probability (also called Experimental Probability) is the probability of an event based on actual experiments or observations โ€” not on theoretical assumptions.

It is calculated by performing an experiment a number of times and recording how often a particular event occurs. The more trials performed, the more reliable the result.

Key idea: As the number of trials increases, the empirical probability gets closer and closer to the theoretical (classical) probability.
๐Ÿ“Œ Example
A coin is tossed 50 times. Heads appeared 28 times.
โˆด Empirical P(Heads) = 2850 = 0.56

(Classical P(Heads) = ยฝ = 0.5 โ€” close but not exact because we only did 50 trials)
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Understanding the Formula
P(E) = fn   where:
P(E)
Empirical probability of event E
Value between 0 and 1
f (frequency)
Number of times the event actually occurred in the experiment
Also written as n(E)
n (total trials)
Total number of times the experiment was performed
Also written as n(T)
Event (E)
The specific outcome we are interested in finding the probability of
e.g. "getting Heads", "rolling 4"
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Classical vs Empirical Probability
Empirical Probability Classical Probability
Based on actual experiments Based on theoretical assumptions
Requires conducting trials Calculated without doing experiments
P(E) = frequency รท total trials P(E) = favourable รท total outcomes
Changes with each new experiment Always the same for a fair experiment
Example: Coin tossed 100 times; Heads came 47 times โ†’ P = 47/100 Example: Fair coin โ†’ P(Heads) = 1/2 always
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Key Properties of Empirical Probability
0 โ‰ค P(E) โ‰ค 1
Empirical probability always lies between 0 and 1
Sum = 1
The sum of empirical probabilities of all outcomes = 1
More trials โ†’ More accurate
As n increases, empirical probability approaches classical probability
P(ฤ’) = 1 โˆ’ P(E)
Probability of NOT getting E equals 1 minus probability of E
Remember: Empirical Probability = Experimental Probability.
Formula: P(E) = Number of times E occurredTotal number of trials

It is always between 0 and 1, i.e. 0 โ‰ค P(E) โ‰ค 1.