# Theoretical and Experimental Probability

## Key Questions

• Theoretical Probability

Assume that each outcome is equally likely to occur.

Let $S$ be a sample space (the set of all outcomes), and let $E$ be an event (a subset of $S$).

The probability of the event $E$ can be found by

$P \left(E\right) = \frac{n \left(E\right)}{n \left(S\right)}$,

where $n \left(E\right)$ and $n \left(S\right)$ denote the number of outcomes in $E$ and the number of outcomes in $S$, respectively.

Example

What is the probability of rolling a multiple of 3 when you roll a standard die once?

Since all outcomes are 1 through 6, we have the sample space

$S = \left\{1 , 2 , 3 , 4 , 5\right\}$

Since all multiple of 3 are 3 and 6, we have the event

$E = \left\{3 , 6\right\}$

Hence, the probability of rolling a multiple of 3 is

$P \left(E\right) = \frac{n \left(E\right)}{n \left(S\right)} = \frac{2}{6} = \frac{1}{3}$

I hope that this was helpful.

See below.

#### Explanation:

Theoretical probability is based on prior knowledge with math. In contrast, experimental probabilty is the probability resulting from an experiment.

For instance, if you are asking for the theoretical probabilty of a dice landing on $1$ in $4$ turns, you would say $\frac{1}{6}$. But if your looking for the experimental probabilty, the answer might be different based on your experiment.

Hope it helps :)

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