14. Discuss relationship between the concept of relative frequency and probability. Note. How defining axioms of probability do correspond to actual properties of the relative frequency

If an action is repeated n times and a certain event occurred b times then the ratio b/n is called the relative frequency. Where as theoretical probability it is used to determine the number of ways that the event can occur if an experiment is repeated a large number of times. The relative frequency is an estimate of the probability of an event.

Probability is nothing else than ratio. We will write the probability of A as P(A) or Pr(A). So what do we mean by probability? A random experiment cannot be predicted in the short run. For example, if we roll a die we cannot tell which number will be rolled. However, we can say a lot about the long run frequency of each of the outcomes. If we roll the die a million times, we know that about a sixth of the rolls will be a 1. This long term frequency is the probability of the outcome.

Relative frequency shows us, same as the probability, that an event occurs at least 0 times, but never a negative number of times. Because if the event does not happen, that means that the frequency and therefore the relative frequency of that event is 0. This is linked to the first probability axiom. Furthermore, the sum of all relative frequencies is equal to 1 and the second probability axiom tells us that the probability of the entire sample space is 1. The sample space is everything possible for our probability experiment and that there are no events outside of the sample space. Clearly, these two are linked. Finally, we can see in the third axiom that the probability of the sum of two events is equal to the sum of their probabilities. This tells us that those events are mutually exclusive, meaning that they have an empty intersection. This axiom corresponds to the third formal property of the relative frequency.

There are three axioms of probability, or rules that probability values must follow and works for relative frequency too .

  1. P(S) = 1. The sample space is the set of all possible outcomes. Therefore, some outcome within S occurs each time, so its relative frequency must be 100%, or 1.
  2. P(A) ≥ 0. The relative frequency must be equal to or greater than 0 since an event cannot occur a negative amount.
  3. If A and B are disjoint events, meaning they share no outcomes(A ∩ B = ∅) , then P(A ∩ B) =0 and P(A U B) = P(A) + P(B).