# Theoretical Probability vs. Experimental Probability: What's the Difference?

Edited by Harlon Moss || By Janet White || Published on November 27, 2023
Theoretical Probability is based on possible outcomes, while Experimental Probability is based on actual trials.

## Key Differences

Theoretical Probability is the likelihood of an event occurring based on all possible outcomes. In contrast, Experimental Probability is derived from the number of times an event occurs over the number of trials conducted.
In Theoretical Probability, calculations are made by assuming each outcome in an experiment is equally likely. Experimental Probability, on the other hand, uses actual data from performing the experiment to determine the probability of an event.
Theoretical Probability is often employed when events are purely random, and all outcomes are known beforehand. However, Experimental Probability comes into play when actual experiments or trials are conducted, and the outcomes are recorded.
In situations where real-life experiments are impractical or impossible, Theoretical Probability is a valuable tool. Experimental Probability, on the contrary, provides insights from real-world data, making it more empirical.
It's worth noting that while Theoretical Probability gives an expected probability, Experimental Probability might differ from this expectation because of the randomness and variations in real-life trials.

## Comparison Chart

### Basis of Calculation

Based on possible outcomes of a random experiment.
Based on actual outcomes of trials conducted.

### Nature

Purely analytical and assumes each outcome is equally likely.
Empirical, relies on real-world data.

### Usage

When real-life trials are impractical or impossible.
When experiments can be performed to gather data.

### Variation with Trials

Remains constant as it's based on inherent characteristics of the event.
May vary with more trials as it depends on actual outcomes.

### Accuracy

Provides expected probability.
Might differ from theoretical due to randomness in real-life situations.

## Theoretical Probability and Experimental Probability Definitions

#### Theoretical Probability

It's independent of empirical results.
Regardless of past events, the Theoretical Probability of a future event remains constant.

#### Experimental Probability

Relies on actual data from performed experiments.
In a game, a player hits 20 shots out of 50 attempts; the Experimental Probability of hitting a shot is 20/50.

#### Theoretical Probability

Assumes all outcomes are equally likely.
In a deck of cards, the Theoretical Probability of drawing an ace is 4/52.

#### Experimental Probability

Might deviate from Theoretical Probability.
Even if a die is expected to land on a 2 with a 1/6 probability, an Experimental Probability might show different results after several rolls.

#### Theoretical Probability

Represents an ideal situation in probability.
The Theoretical Probability of choosing a red marble from a bag with 5 reds and 5 greens is 1/2.

#### Experimental Probability

Experimental Probability is based on observed results of trials.
After flipping a coin 100 times and getting 60 heads, the Experimental Probability of heads is 60/100.

#### Theoretical Probability

It's derived without actual experimentation.
The Theoretical Probability of rolling a 3 on a fair six-sided die is 1/6.

#### Experimental Probability

Can change with more trials due to variations in outcomes.
After an additional 50 coin flips with 25 heads, the Experimental Probability changes based on the total outcomes.

#### Theoretical Probability

Theoretical Probability calculates chances based on all known outcomes.
The Theoretical Probability of flipping heads on a coin is 1/2.

#### Experimental Probability

Provides empirical insights into the likelihood of events.
After drawing 5 red cards from a shuffled deck in 10 tries, the Experimental Probability of drawing red is 5/10.

## FAQs

#### How is Experimental Probability determined?

It's calculated from actual results of experiments or trials.

#### Does Theoretical Probability change with more trials?

No, it remains constant based on inherent event characteristics.

#### Why might we use Theoretical Probability?

It's used when real-life trials are impractical or when outcomes are inherently known.

#### Does Experimental Probability always get closer to Theoretical Probability with more trials?

Typically, with a large number of trials, the two probabilities converge due to the Law of Large Numbers.

#### Is Experimental Probability more accurate than Theoretical Probability?

Not necessarily. Both have their applications, but Experimental Probability gives insight into real-world occurrences.

#### What is Theoretical Probability?

Theoretical Probability estimates the chance of an event based on possible outcomes.

#### Can Experimental Probability differ from Theoretical Probability?

Yes, due to randomness and variations in real-life trials.

#### Can Theoretical Probability be zero?

Yes, if an event is impossible.

#### Can the two probabilities be equal?

Yes, especially in controlled conditions where experimental outcomes match expected outcomes.

#### Why might Experimental Probability differ from expectations?

Real-world randomness, limited trials, or experimental error can cause deviations.

#### What's the maximum value for both probabilities?

Both probabilities range between 0 and 1, inclusive.

#### Can we predict future results with Experimental Probability?

It can give insights but isn't definitive due to inherent randomness in future trials.

#### Is Theoretical Probability always a rational number?

Not necessarily, though it's often expressed as a fraction.

#### Can Experimental Probability be used to validate theories?

Yes, if experimental outcomes consistently match theoretical expectations, it can support the validity of a theory.

#### How do we calculate Theoretical Probability?

Divide the number of favorable outcomes by the total number of possible outcomes.

#### What's an example of Experimental Probability?

If it rains 10 days out of 30 in a month, the Experimental Probability of rain on a given day that month is 10/30.

#### Do casinos use Theoretical or Experimental Probability?

Casinos rely on Theoretical Probability but monitor Experimental Probability to detect anomalies.

#### Why study both types of probabilities?

Understanding both gives a comprehensive view of how events might occur theoretically versus in practice.

#### How is Theoretical Probability useful in games of chance?

It provides the expected odds of specific outcomes.

#### Can experimental errors impact Experimental Probability?

Yes, errors in data collection or trial conduction can skew results.