# 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.

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