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

By Tayyaba Rehman — Published on November 26, 2023
Theoretical Probability is the likelihood based on mathematical reasoning, while Experimental Probability is based on actual trials or experiments. Theoretical is ideal; Experimental is observed.

## Key Differences

Theoretical Probability is essentially the chance of an event occurring when all outcomes are equally likely. It's a ratio of the number of favorable outcomes to the number of possible outcomes. Experimental Probability, on the other hand, is determined by conducting experiments or trials and observing the number of times an event occurs.
In Theoretical Probability, one doesn't need to conduct an experiment; it's purely based on mathematical principles. For example, when flipping a coin, the theoretical probability of getting heads is 0.5. Experimental Probability, conversely, would require one to flip the coin several times and record how often heads appear to determine its probability.
Theoretical Probability often provides an idealized perspective on likelihood, based on presumptions or mathematical structure. For instance, a fair dice's theoretical probability to land on any given side is 1/6. Experimental Probability provides real-world data. If one rolls a dice 600 times, maybe it lands on a '3' 110 times, giving an experimental probability of 110/600.
While Theoretical Probability operates on the assumption of ideal conditions (like a fair coin or dice), Experimental Probability works with actual conditions, which might include imperfections or biases. For instance, a coin might be slightly weighted towards tails, impacting the Experimental Probability but not the Theoretical.
Ultimately, Theoretical Probability gives a baseline expectation, rooted in mathematics and logic. Experimental Probability provides a measure of what actually happens, based on real-life observations and tests, and might differ from theoretical predictions due to various factors.

## Comparison Chart

### Basis

Mathematical reasoning
Actual trials or experiments

### Need for experiments

No experiments needed
Requires experiments

### Predictability

Based on idealized conditions
Based on observed outcomes

### Factors considered

Assumes all outcomes are equally likely
Considers real-world conditions, biases, imperfections

### Potential for variance

Doesn't vary; it's a fixed value
Can vary based on different trials

## Compare with Definitions

#### Theoretical Probability

Likelihood of an event based on mathematical analysis.
The Theoretical Probability of rolling a 4 on a fair dice is 1/6.

#### Experimental Probability

Likelihood calculated from actual data and occurrences.
In 100 coin flips, if heads appears 53 times, the Experimental Probability of heads is 53/100.

#### Theoretical Probability

Chance of an occurrence based on logical reasoning.
The Theoretical Probability of getting tails in a coin flip is 50%.

#### Experimental Probability

Likelihood grounded in practical observation.
After shooting 100 free throws and making 80, a player's Experimental Probability of making a shot is 80%.

#### Theoretical Probability

Probability derived from known facts and assumptions.
Given all socks in a drawer are red or blue, the Theoretical Probability of picking a red sock is the ratio of red socks to total socks.

#### Experimental Probability

Probability based on observed outcomes from trials.
If a dice lands on 5, twenty times out of 100 rolls, the Experimental Probability of rolling a 5 is 20%.

#### Theoretical Probability

A measure of likelihood based on inherent structure or symmetry.
In a lottery with 1000 tickets, the Theoretical Probability of winning with one ticket is 1/1000.

#### Experimental Probability

A measure of chance based on empirical evidence.
If it rained 15 days out of a 30-day month, the Experimental Probability of rain on any given day that month was 50%.

#### Theoretical Probability

Probability calculated without the need for experimentation.
The Theoretical Probability of drawing an ace from a standard deck is 4/52.

#### Experimental Probability

Probability derived from real-world tests or experiments.
After surveying 500 people, the Experimental Probability of a preference for chocolate over vanilla was determined from the responses.

## Common Curiosities

#### How is Experimental Probability calculated?

It's the ratio of the number of times an event occurs to the total number of trials.

#### Does Experimental Probability always match Theoretical Probability?

No, Experimental Probability is based on actual trials and might vary from the theoretical due to various factors.

#### Can Theoretical Probability change?

Typically, it remains constant based on logical reasoning, while Experimental can vary with different trials.

#### Is Theoretical Probability always accurate?

It provides a baseline expectation, but real-world outcomes (Experimental Probability) might differ due to biases or imperfections.

#### Why might Theoretical and Experimental Probabilities differ?

Factors like experimental biases, imperfect tools, or random chance can cause discrepancies.

#### What's more reliable, Theoretical or Experimental Probability?

Both have value. Theoretical gives idealized expectations, while Experimental provides real-world data.

#### How is Theoretical Probability determined?

It's determined using mathematical reasoning without actual experimentation.

#### Do we need experiments for Theoretical Probability?

No, it's derived from mathematical or logical analysis.

#### Are there situations where only one type of probability is relevant?

In purely theoretical scenarios, only Theoretical Probability might be considered. In real-world testing, Experimental becomes more relevant.

#### Can Experimental Probability be predicted?

It's derived from actual trials, so while it can be estimated using Theoretical Probability, it can't be precisely predicted.

#### Does a higher number of trials improve Experimental Probability's accuracy?

Yes, more trials often lead to results closer to the Theoretical Probability.

#### Do biases affect Theoretical Probability?

Theoretical Probability is based on logic and mathematics. However, if there's an error in reasoning or incomplete information, biases might be introduced.

#### Can the environment affect Experimental Probability?

Yes, factors like temperature, human error, or equipment can impact outcomes.

#### Is Theoretical Probability based on assumptions?

It's based on known facts and logical reasoning, often assuming ideal conditions.

#### How do casinos use these probabilities?

Casinos understand both types, setting games with favorable Theoretical Probabilities, but actual outcomes (Experimental) may vary in the short term.