# 7.1 Probability: part 2

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## Relative frequency vs. Probability

There are two approaches to determining the probability associated with any particular event of a random experiment:

1. determining the total number of possible outcomes and calculating the probability of each outcome using the definition of probability
2. performing the experiment and calculating the relative frequency of each outcome

Relative frequency is defined as the number of times an event happens in a statistical experiment divided by the number of trials conducted.

It takes a very large number of trials before the relative frequency of obtaining a head on a toss of a coin approaches theprobability of obtaining a head on a toss of a coin (in fact, if the probability of an event occurring is something other than 1, i.e. it is always true, or 0, i.e. it is never true, then probability is only absolutely accurate when an infinite number of trials is conducted). For example, the data in [link] represent the outcomes of repeating 100 trials of a statistical experiment 100 times, i.e. tossing a coin 100 times.

 H T T H H T H H H H H H H H T H H T T T T T H T T H T H T H H H T T H T T H T T T H H H T T H T T H H T T T T H T T H H T T H T T H T T H T H T T H T T T T H T T H T T H H H T H T T T T H H T T T H T

The following two worked examples show that the relative frequency of an event is not necessarily equal to the probability ofthe same event. Relative frequency should therefore be seen as an approximation to probability.

Determine the relative frequencies associated with each outcome of the statistical experiment detailed in [link] .

1. There are two unique outcomes: H and T.

2.  Outcome Frequency H 44 T 56
3. The statistical experiment of tossing the coin was performed 100 times. Therefore, there were 100 trials, in total.

4. $\begin{array}{ccc}\hfill \mathrm{Probability of H}& =& \frac{\mathrm{frequency of outcome}}{\mathrm{number of trials}}\hfill \\ & =& \frac{44}{100}\hfill \\ & =& 0,44\hfill \\ \\ \hfill \mathrm{Relative Frequency of T}& =& \frac{\mathrm{frequency of outcome}}{\mathrm{number of trials}}\hfill \\ & =& \frac{56}{100}\hfill \\ & =& 0,56\hfill \end{array}$

The relative frequency of the coin landing heads-up is 0,44 and the relative frequency of the coin landing tails-up is 0,56.

Determine the probability associated with an evenly weighted coin landing on either of its faces.

1. There are two unique outcomes: H and T.

2. There are two possible outcomes.

3. $\begin{array}{ccc}\hfill \mathrm{Relative Frequency of H}& =& \frac{\mathrm{number of favourable outcomes}}{\mathrm{total number of outcomes}}\hfill \\ & =& \frac{1}{2}\hfill \\ & =& 0,5\hfill \\ \\ \hfill \mathrm{Relative Frequency of T}& =& \frac{\mathrm{number of favourable outcomes}}{\mathrm{total number of outcomes}}\hfill \\ & =& \frac{1}{2}\hfill \\ & =& 0,5\hfill \end{array}$

The probability of an evenly weighted coin landing on either face is $0,5$ .

## Project idea

Perform an experiment to show that as the number of trials increases, the relative frequency approaches the probability of a cointoss. Perform 10, 20, 50, 100, 200 trials of tossing a coin.

## Interpretation of probability values - (not in caps, included for completeness)

The probability of an event is generally represented as a real number between 0 and 1, inclusive. An impossible event has aprobability of exactly 0, and a certain event has a probability of 1, but the converses are not always true: probability 0 events are not always impossible,nor probability 1 events certain. There is a rather subtle distinction between "certain" and "probability 1".

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