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In statistics, it is very common to be faced with situations in which you want to calculate the union probability of several different events. For example, the owner of a candy store may be interested in determining what is the probability that the next child who enters his store will purchase a white chocolate bar or a milk chocolate bar. In this case, we want to determine the probability of one of two possible events happening, which, according to set theory, is the union probability of both events, or P(AUB).
In the case described, the calculation of this probability simply consists of the sum of the individual probabilities, minus the probability of the intersection between both events, that is:
The reason the intersection probability must be subtracted is that by adding the probabilities of both events, any intersection is being counted twice. This is a relatively simple process to understand. However, it may also happen that we want to determine the union probability not of two, but of three or more events. What should be done in such cases? In the next section we will look at a simple way to determine the formula to apply in the three-event and four-event cases, and then we will use these results, along with the above formula, to generalize the determination of the union probability for any number of events. events.
Basics review
To understand the process of calculating union probabilities, it is necessary to briefly recall some important terms that will be used later:
experiment . In probability, an experiment is any process that can be repeated multiple times and always produces a result. Each experiment is associated with a certain set of possible outcomes that will always be the same.
Result . We will call the consequence of an experiment a result, such as the particular face that comes out when throwing a dice.
Sample space (S) . The set of all possible outcomes of an experiment.
event . Any set of possible outcomes.
Venn diagram . Graphical representation that shows the relationships between sets of events and between the probability of events in an experiment.
The union probability of three events
Suppose we carry out an experiment and we want to determine the probability of one of 3**3three different events occurring, which may or may not occur simultaneously. We will call these three events A, B, and C.
In these cases, several different situations can occur. For example, it may happen that none of the events share results with any other, in which case we say that the events are mutually exclusive, which is exemplified in the following Venn diagram:
Circles A, B and C represent the three events and enclose a set of results within the sample space, which is the gray rectangle identified with the letter S. In these cases, the union probability is simply given by the sum of the probabilities of each separate event:
On the other hand, one of the events may also share results with one of the other two events, or even with both. This is illustrated in a Venn diagram as areas that intersect each other.
In these cases, the sum of the probabilities takes some outcomes into account more than once, so it is necessary to subtract these probabilities that have been overcounted. That is, we must subtract the probability of the intersection between each pair of events. However, in cases where there are outcomes present in all three events (such as those at the center of the Venn diagram above), subtraction of the intersections of the pairs removes the contribution of the central area at which the pairs intersect. three events. For this reason, we must add again this small area that corresponds to the probability of intersection of A, B and C.
Finally, the union probability of the three events is:
NOTE: Although this expression was stated for the particular case where the three events intersect each other, this is the more general form of the three-event case since it can be converted to the union probability of any set of three events, whether they intersect or not. For example, in the case of mutually exclusive events, all intersection probabilities are zero, so the expression reduces to the sum of the individual probabilities shown at the beginning of this section.
The union probability of four events
Suppose now that we carry out a new experiment and are interested in the probability of union between four events: A, B, C and D. The most general case is that they can all intersect each other, as shown in the following diagram:
In this case, the sum of the four simple probabilities counts four times the probability of the outcomes contained in area I, three times those of areas II, III, IV, and V, and twice those of areas VI, VII, VIII, and IX. To correct this, we must first subtract the intersection probabilities of all pairs (A and B, A and C, A and D, B and C, B and D, and C and D). This, in turn, subtracts the regions of intersection of each group of three (ABC, ABD, ACD, and BCD) too many times, so these areas must be added again, and so on until all areas are counted once. once.
The result for the case of four events, whether mutually exclusive or not, is:
Union probability of more than four events
Up to this point we can already detect a pattern between the formulas for the union probabilities of two, three and four events. They all start with the sum of the simple probabilities, then subtract the intersection probabilities between all possible pairs of events, then add the intersection probabilities of each possible group of three events, and so on, alternately adding and subtracting the intersections. between more and more events until we reach the intersection of all events. For an even number of events, this last intersection is always negative (subtracted) while, for an odd number of events, it is always positive (added).
References
- Arrizabalaga R., M. (2015, September). PROBABILITY THEORY . AUTONOMOUS MEXICO STATE UNIVERSITY. https://core.ac.uk/download/pdf/55528069.pdf
- DeVore, J. (2002). Probability and Statistics for Engineering and Sciences (5th ed.). Thomson International.
- Manuel, M. (2020, July 1). Probability of the Union of Events . Easy Maths. https://lasmatesfaciles.com/2020/06/29/probabilidad-de-la-union-de-sucesos/
- Marta, M. (2021, March 27). Union of events or occurrences . Superprof. https://www.superprof.es/apuntes/escolar/matematicas/probabilidades/combinatoria/union-de-sucesos.html
