Relative Frequency

The term relative frequency is used for the ratio of the observed frequency of an outcome and the total frequency of the random experiment. Suppose a random experiment is repeated $$N$$ times and an outcome is observed $$f$$ times. In this case, the ratio $$\frac{f}{N}$$ is called the relative frequency of the outcome which has been observed $$f$$ times. Some examples of relative frequencies are:

  • We select bulbs from a certain big lot to examine whether they are good or defective. We take $$100$$ such bulbs and examine them. Sixty bulbs are found to be defective. The symbol $$N$$ may be used for $$100$$ and the symbol $$f$$ may be used for the observed frequency, which is $$60$$. Thus the \[{\text{Relative frequency}} = \frac{f}{N} = \frac{{60}}{{100}} = 0.6\]
  • We are interested in knowing whether a coin is unbiased (true) or not. We toss the coin $$200$$ times and note the number of heads. In $$200$$ tosses, the number of heads is $$110$$. The relative frequency of this experiment for the number of heads is $$\frac{{110}}{{200}}$$, which is not $$\frac{1}{2}$$. As we shall see later, the probability of heads is usually written as $$\frac{1}{2}$$. It is just an assumption and, of course, a big assumption. If we repeat the same experiment again, the number of heads may be less than or more than $$110$$ as was observed in the first experiment. This is what happens in random experiments.
  • A die is thrown and we are interested in the ace (face 1). We throw the die $$60$$ times and the ace is observed $$12$$ times. Thus the relative frequency of the ace is $$\frac{{12}}{{60}} = \frac{1}{5}$$. For an ideal die one should expect that the number of aces would be $$\frac{{60}}{6} = 10$$. At some later stage we would like to know more about the ratio $$\frac{{12}}{{60}}$$. This ratio is not constant. The next random experiment with the same die may produce a completely different result.