Studying the theory of probability begins with solving problems of addition and multiplication of probabilities. It is worth mentioning right away that when mastering this field of knowledge, a student may encounter a problem: if physical or chemical processes can be represented visually and understood empirically, then the level of mathematical abstraction is very high, and understanding here comes only with experience.
However, the game is worth the candle, because the formulas - both considered in this article and more complex ones - are used everywhere today and may well be useful in work.
Origin
Oddly enough, the impetus for the development of this section of mathematics was … gambling. Indeed, dice, coin toss, poker, roulette are typical examples that use addition and multiplication of probabilities. On the example of tasks in any textbook, this can be seen clearly. People were interested in learning how to increase their chances of winning, and I must say, some succeeded in this.
For example, already in the 21st century, one person, whose name we will not disclose,used this knowledge accumulated over the centuries to literally “cleanse” the casino, winning several tens of millions of dollars at roulette.
However, despite the increased interest in the subject, it was not until the 20th century that a theoretical framework was developed that made the “theorver” a full-fledged component of mathematics. Today, in almost any science, you can find calculations using probabilistic methods.
Applicability
An important point when using formulas of addition and multiplication of probabilities, conditional probability is the satisfiability of the central limit theorem. Otherwise, although it may not be realized by the student, all calculations, no matter how plausible they may seem, will be incorrect.
Yes, the highly motivated learner is tempted to use new knowledge at every opportunity. But in this case, one should slow down a little and strictly outline the scope of applicability.
Probability theory deals with random events, which in empirical terms are the results of experiments: we can roll a six-sided die, draw a card from a deck, predict the number of defective parts in a batch. However, in some questions it is categorically impossible to use formulas from this section of mathematics. We will discuss the features of considering the probabilities of an event, the theorems of addition and multiplication of events at the end of the article, but for now let's turn to examples.
Basic concepts
A random event means some process or result that may or may not appearas a result of the experiment. For example, we toss a sandwich - it can fall butter up or butter down. Either of the two outcomes will be random, and we do not know in advance which of them will take place.
When studying addition and multiplication of probabilities, we need two more concepts.
Joint events are those events, the occurrence of one of which does not exclude the occurrence of the other. Let's say two people shoot at a target at the same time. If one of them fires a successful shot, it won't affect the other's ability to hit or miss.
Inconsistent will be such events, the occurrence of which is simultaneously impossible. For example, by pulling only one ball out of the box, you cannot get both blue and red at once.
Designation
The concept of probability is denoted by the Latin capital letter P. Next in brackets are arguments denoting some events.
In the formulas of the addition theorem, conditional probability, multiplication theorem, you will see expressions in brackets, for example: A+B, AB or A|B. They will be calculated in various ways, we will now turn to them.
Addition
Let's consider cases where addition and multiplication formulas are used.
For incompatible events, the simplest addition formula is relevant: the probability of any of the random outcomes will be equal to the sum of the probabilities of each of these outcomes.
Suppose there is a box with 2 blue, 3 red and 5 yellow balloons. There are 10 items in total in the box. What is the percentage of the truth of the statement that we will draw a blue or red ball? It will be equal to 2/10 + 3/10, i.e. fifty percent.
In the case of incompatible events, the formula becomes more complicated, since an additional term is added. We will return to it in one paragraph, after considering one more formula.
Multiplication
Addition and multiplication of probabilities of independent events are used in different cases. If, according to the condition of the experiment, we are satisfied with either of the two possible outcomes, we will calculate the sum; if we want to get two certain outcomes one after the other, we will resort to using a different formula.
Returning to the example from the previous section, we want to draw the blue ball first and then the red one. The first number we know is 2/10. What happens next? There are 9 balls left, there are still the same number of red ones - three pieces. According to the calculations, you get 3/9 or 1/3. But what to do with two numbers now? The correct answer is to multiply to get 2/30.
Joint Events
Now we can revisit the sum formula for joint events. Why are we digressing from the topic? To learn how probabilities are multiplied. Now this knowledge will come in handy.
We already know what the first two terms will be (the same as in the addition formula considered earlier), now we need to subtractthe product of probabilities that we have just learned to calculate. For clarity, we write the formula: P (A + B) u003d P (A) + P (B) - P (AB). It turns out that both addition and multiplication of probabilities are used in one expression.
Let's say we have to solve either of the two problems to get credit. We can solve the first one with a probability of 0.3, and the second - 0.6. Solution: 0.3 + 0.6 - 0.18=0.72. Note that simply summing the numbers here will not be enough.
Conditional Probability
Finally, there is the concept of conditional probability, the arguments of which are indicated in brackets and separated by a vertical bar. The entry P(A|B) reads as follows: "probability of event A given event B".
Let's look at an example: a friend gives you some device, let it be a phone. It can be broken (20%) or good (80%). You are able to repair any device that falls into your hands with a probability of 0.4 or you are not able to do it (0.6). Finally, if the device is in working condition, you can reach the right person with a probability of 0.7.
It's easy to see how conditional probability works in this case: you can't get through to a person if the phone is broken, and if it's good, you don't need to fix it. Thus, in order to get any results on the "second level", you need to know what event was executed on the first one.
Calculations
Let's consider examples of solving problems on addition and multiplication of probabilities, using the data from the previous paragraph.
First, let's find the probability that yourepair the device given to you. To do this, firstly, it must be faulty, and secondly, you must cope with the repair. This is a typical multiplication problem: we get 0.20.4=0.08.
What is the probability that you will immediately get through to the right person? Easier than simple: 0.80.7=0.56. In this case, you found that the phone is working and successfully made a call.
Finally, consider this scenario: you received a broken phone, fixed it, then dialed the number, and the person on the opposite end answered the phone. Here, the multiplication of three components is already required: 0, 20, 40, 7=0, 056.
And what if you have two non-working phones at once? How likely are you to fix at least one of them? This is a problem of addition and multiplication of probabilities, since joint events are used. Solution: 0, 4 + 0, 4 – 0, 40, 4=0, 8 – 0, 16=0, 64.
Careful use
As mentioned at the beginning of the article, the use of probability theory should be deliberate and conscious.
The larger the series of experiments, the closer the theoretically predicted value approaches the practical one. For example, we are tossing a coin. Theoretically, knowing about the existence of formulas for addition and multiplication of probabilities, we can predict how many times heads and tails will fall out if we conduct the experiment 10 times. We did an experiment andCoincidentally, the ratio of the dropped sides was 3 to 7. But if you conduct a series of 100, 1000 or more attempts, it turns out that the distribution graph is getting closer and closer to the theoretical one: 44 to 56, 482 to 518 and so on.
Now imagine that this experiment is carried out not with a coin, but with the production of some new chemical substance, the probability of which we do not know. We would run 10 experiments and, without getting a successful result, we could generalize: "the substance cannot be obtained." But who knows, if we made the eleventh attempt, would we have reached the goal or not?
So if you're going into the unknown, the unexplored realm, the theory of probability may not apply. Each subsequent attempt in this case may be successful and generalizations like "X does not exist" or "X is impossible" will be premature.
Closing word
So we've looked at two types of addition, multiplication and conditional probabilities. With further study of this area, it is necessary to learn to distinguish situations when each specific formula is used. In addition, you need to understand whether probabilistic methods are generally applicable to solving your problem.
If you practice, after a while you will begin to carry out all the required operations exclusively in your mind. For those who are fond of card games, this skill can be consideredextremely valuable - you will significantly increase your chances of winning, just by calculating the probability of a particular card or suit falling out. However, the acquired knowledge can easily be applied in other areas of activity.
