1.4.3 Probability Measure

A probability measure is a function that assigns a probability to each event within the event space, the probability is a real number between 0 and 1. This can be denoted as:

        

 

As  is a function, the input to this function is an event and the output is the probability of that event. A completely impossible event has a probability of 0, whereas an event that is certain to occur has a probability of 1. However, apart from the assignment of 1 to the sample space set and 0 to the empty set, the probability measure function doesn't provide specific guidance or rules for determining the initial values (probabilities) that should be assigned to different events. For guidance with that, we need to turn to the interpretations of probability, like classical, empirical, or subjective (discussed earlier in section 2).

To understand it with an analogy: Think of the probability measure as a “black box” into which you input an event, and it outputs a number between 0 and 1. However, this definition doesn’t tell you the internal workings of this “black box” for any specific random experiment.

For instance, in the context of rolling a fair six-sided die:

• The probability measure tells you that the probability of getting any one specific number (say a 2) lies between 0 and 1.
• The specific interpretation of probability (like classical in this case) will tell you that the probability is .

Another example: In a subjective interpretation of probability, different people might assign slightly different probabilities to the same event based on their beliefs or information. The probability measure accommodates all these values as long as they are between 0 and 1, but it doesn't dictate which specific value should be assigned.

Example 1.23. Continuing with the experiment of “tossing a fair coin” from the previous section, here we can apply the classical interpretation of probability since all the coin’s outcomes are distinct and equally likely to occur. Thus, the probability of event A can be calculated using the formula:

        

•  is the number of outcomes that can result in the occurrence of .
•  is the total number of possible outcomes in the sample space.

When the event is the sample space itself, its set contains “heads” and “tails”, which can be represented as {, }. Consequently, there are two outcomes () that can result in the occurrence of the “sample space” event. Since a coin has two possible outcomes (),  is also two. Thus, we can establish a mapping between all the events in the event space  and their corresponding probabilities as follows:

        

        

        

        

Fig 1.12 shows the probability measure mapping the empty set to zero, the sample space to 1 and any other event to some number in [0,1].

Probability space gives us a framework to transition from a real-world experiment to defining a comprehensive sample space, constructing an event space to categorize a specific event of interest, and finally feeding this event to the probability measure function to determine its probability by incorporating the relevant probability interpretation. However, the probability measure function must follow certain rules to ensure that the assigned probabilities are consistent, logical and mathematically meaningful. The subsequent section explores these rules in depth.