Human life is full of uncertainty and chance, which, as it often is, determines the course of being. Unplanned encounters, abrupt changes, accidents, injuries, or luck can significantly affect the quality of life, and even if an individual seeks only authorized and pre-modeled conditions, randomness cannot be eliminated. Strictly speaking, mathematical probability refers to the degree of possibility of a circumstance occurring whose favorable factors outweigh the opposite grounds (Siegmund). In other words, a person can expect to get into an accident while driving a car, but the probability of being killed in a two-car collision, if a person is in an apartment, is practically zero.

In fact, the phenomenon of mathematical probability is a constant companion in carrying out any action. In the seventeenth century, two mathematicians, Pierre Fermat and Blaise Pascal, noticed some of the regularities associated with the fallout of a dice game: the result of their observations was the formalized theory of probability (Boyer). Fundamentally, this theory operates in terms of favorable random events and the number of all possible outcomes, which is characterized by the following formulaic expression:

For example, tails’ probability on a two-sided coin is 50% since only one side satisfies the condition. However, an important question is how to calculate the probability of the sum of a finite number of incompatible events (More). More specifically, if there were three 25- cent coins, four 50-cent coins, and three dimes among the ten coins, then determining the probability of taking either a dime or a 25-cent is more complicated. To do this, it is necessary

to use the theorem about the sum of probabilities of incompatible events:

In this case, the probability of getting either a dime or 25 cents out of a set of coins is 60%:

It is not difficult to prove this theorem: it is necessary to consider the sum of two ^{incompatible events, A}1 ^{and A}2^{. Thus, if event A}1 ^{favors n}1 ^{elementary outcomes, and event} ^{A}2^{, n}2^{, then given their incompatibility, the probability of event A}1^{+A}2 ^{fits n}1^{+n}2 ^{elementary} outcomes of the total number k. It follows that:

The theorem is proven. To summarize, it is worth acknowledging that probabilities occur in real life, and many unplanned events are random. It is possible to calculate the percentage of probability of such outcomes using general rules and theorems. For incompatible events in which alternative outcomes are possible, it is appropriate to use the probability sum theorem, illustrated in this essay.

## Works Cited

Boyer, Carl. “Pierre de Fermat.” *Britannica*, 2021, Web.

More, Hemant. “Events and Their Types.” *The Fact Factor*, 2020, Web.

Siegmund, David. “Probability Theory.” *Britannica*, 2020, Web.