Tabla de Contenidos
The addition rules in probability and statistics refer to the different ways in which we can combine known probabilities of two or more different events to determine the probability of new events formed by the union of those events .
In statistics and probability, we often know the probability that certain events (for example, events A and B) will occur separately but not the probability that they will occur at the same time or that one or the other will occur. This is where the addition rules come in handy.
For example: we can know the probability of rolling a six when rolling two dice, call it P(rolling 6), and the probability of both dice landing on even numbers, call it P(even numbers).
This is relatively easy. But sometimes we are interested in determining the probability that, when two dice are rolled, they both come up with an even number or that they add up to six. In statistical notation and in group theory, this “or” is represented with the symbol U that indicates the union of two events and in this case, this probability would be represented as follows:
These types of probabilities can be calculated from the individual probabilities and some additional data by means of the addition rules.
It should be noted that which addition rule we should use in each case depends both on the number of events we are considering and on whether or not these events are mutually exclusive. The addition rules for some simple cases are described below.
Case 1: Addition rule for disjoint or mutually exclusive events
Two events are called mutually exclusive when the occurrence of one of them excludes the possibility of the other occurring. That is, they are events that cannot occur at the same time. For example, when throwing a dice, the one that the result in which 4 comes up excludes that any of the other 5 possible results have come up.
If we consider two or more events (A, B, C…) mutually exclusive, the union probability simply consists of the sum of the individual probabilities of each of these events. That is, in this case the union probability is given by:
This can be most easily understood by means of a Venn diagram. Here the sample space is represented by a rectangular area; while, the probability of each event is represented by sectors within this larger area. In a Venn diagram, mutually exclusive events are seen as separate areas that neither touch nor overlap.
In this type of diagrams, calculating the union probability consists of obtaining the total area occupied by all the events whose probabilities we are considering. In the case of the previous image, this implies obtaining the total area of sectors A, B and C, that is, the blue area in the following figure.
It is easy to see that if the events are disjoint as in the case of the two images above, the union probability is simply the sum of the three areas.
Example 1: Calculation of the probability of obtaining an even result when rolling a dice
Suppose we roll a dice and want to know the probability of getting an even number. Since the only possible even numbers on a 6-sided die are 2, 4, and 6, then what we really want to know is the probability that the die will land on 2, 4, or 6, since on either of these cases would have fallen into an even number.
The probability of getting any of the 6 heads is 1/6 (as long as it is a fair die). Also, as we saw a moment ago, the three outcomes are mutually exclusive events since, if 2 rolls, 4 or 6 could not have rolled, and so on. Under these conditions, the union probability is given by:
Case 2: Addition rule for two events that are not mutually exclusive
If A and B are events that share outcomes with each other, that is, they can occur at the same time, the events are said not to be mutually exclusive. In this case, the Venn diagram looks like this:
As can be seen, there is a region of the sample space in which both events occur at the same time. If we want to determine the probability of union, that is, P(AUB), we need to find the area indicated in the Venn diagram on the right in the previous figure.
It’s easy to see that in this case, if we just add the areas of A and B, we’ll be counting the common area twice, so we’ll get an area (read, probability) larger than what we want. To correct this excess error, it is only necessary to subtract the area shared by events A and B, which corresponds to the probability of intersection:
This expression for the probability of union also applies to the previous case since, being mutually exclusive, the probability that they occur at the same time (the probability of intersection) is zero.
Example 2: Calculation of the probability of obtaining an even result or obtaining a number less than 4 when rolling a dice
In this case, both events share outcome 2, which is both even and less than 4, so the union probability will be:
Case 3: Addition rule for three events that are not mutually exclusive
Another slightly more complex case is when 3 events occur that are not mutually exclusive, such as the one shown in the following Venn diagram:
In this case, the sum of the three areas counts twice the intersection zones between A and B, between B and C and between C and D, and counts three times the intersection zone of the three events A, B and C. If we do as before and subtract the areas of intersection between each pair of events from the sum of the three areas, we will be subtracting three times the area of the center, so it must be added as the probability of intersection of the three events. Finally, the general addition rule for three non-exclusive events is given by:
As before, this expression is general for any set of three events, whether they are disjoint or not, since, in this case, the intersections will be empty and the result will be the same expression of the first case.
Example 3: Calculation of the probability of getting an even number, a number less than 10 or a prime number on a 20-sided die
In this case, there are three events that share outcomes between and also contain outcomes that are not shared, so the union probability is given by the aforementioned expression.
The probabilities of the individual events are:
Now, the intersection probabilities are:
Now, applying the equation for the union probability:
References
- brilliant. (nd). Probability – Rule of Sum | Brilliant Math & Science Wiki . Retrieved from https://brilliant.org/wiki/probability-rule-of-sum/
- lumen. (nd). Probability Rules | Boundless Statistics . Retrieved from https://courses.lumenlearning.com/boundless-statistics/chapter/probability-rules/#:%7E:text=The%20addition%20rule%20states%20the,probability%20that%20both%20will%20happen .
- MateMobile. (2021, January 1). Rule of the sum or addition of probabilities | matermobile . Retrieved from https://matemovil.com/regla-de-la-suma-o-adicion-de-probabilidades/
- Webster, A. (2001). Statistics Applied to Business and the Economy (Spanish Edition) . Toronto, Canada: Irwin Professional Publishing.
