The study of probability theory begins with the decisiontasks for addition and multiplication of probabilities. It is worth mentioning at once that a student may face a problem while mastering this field of knowledge: if physical or chemical processes can be visualized 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 - are used everywhere today and may well be useful in the work.
Oddly enough, the impetus for the development of thisThe maths section became ... 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 to learn how to increase their chances of winning, and, I must say, some have succeeded in this.
However, despite the increased interest insubject, only to the XX century, a theoretical framework was developed, making the "theorever" a full-fledged component of mathematics. Today, in almost any science, you can find calculations using probabilistic methods.
The important point when using formulasand multiplying probabilities, the conditional probability is the feasibility of the central limit theorem. Otherwise, although it may not be realized by the student, all calculations, no matter how believable they may seem, will be incorrect.
Yes, a highly motivated student is tempted to use new knowledge at every opportunity. But in this case it is necessary to slow down and strictly delineate the scope of applicability.
Probability theory deals with randomevents, which in empirical terms represent the results of experiments: we can roll a cube with six faces, draw a card from the deck, predict the number of defective parts in a game. However, in some questions the use of formulas from this section of mathematics is absolutely impossible. We will discuss the features of considering probabilities of an event, theorems of addition and multiplication of events at the end of the article, but for now let's turn to examples.
By random event is meant somea process or result that may or may not appear as a result of an experiment. For example, we throw up a sandwich - it can drop butter up or butter down. Any of the two outcomes will be random, and we do not know in advance which one will take place.
Joint called such events, the appearanceone of which does not exclude the appearance of the other. Let's say two people simultaneously shoot at a target. If one of them makes a successful shot, it will not affect the ability of the second to hit the bull's eye or miss.
There will be incompatible such events, the appearance of which is at the same time impossible. For example, pulling out of the box only one ball, you can not immediately get both blue and red.
The notion of probability is denoted by the Latin capital letter P. Next, in parentheses are the arguments for some events.
In the formulas of the addition theorem, conditionalprobabilities, multiplication theorems, you will see expressions in parentheses, for example: A + B, AB or A | B. They will be calculated in various ways, we now turn to them.
Consider cases in which the formulas of addition and multiplication of probabilities 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.
In the case of incompatible events, the formula becomes complicated as an additional term is added. We return to it in one paragraph, after considering another formula.
Addition and multiplication of independent probabilitiesevents are used in different cases. If, by 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 another, we will resort to using a different formula.
Returning to the example from the previous section, wewe want to pull out first the blue ball, and then the red one. The first number we know is 2/10. What happens next? There are 9 balls left; there are as many red among them - three pieces. According to calculations, it will be 3/9 or 1/3. But now what to do with two numbers? The correct answer is to multiply to make 2/30.
Now you can again refer to the sum formula for joint events. Why are we distracted from the topic? To find out how probabilities are multiplied. Now this knowledge is useful to us.
Допустим, мы должны решить любую из двух задач, to get credit. We can solve the first with a probability of 0.3, and the second - 0.6. The solution: 0.3 + 0.6 - 0.18 = 0.72. Note, just summing the numbers here will not be enough.
Finally, there is the concept of conditional probability,whose arguments are indicated in parentheses and are separated by a vertical bar. The P (A | B) record is read as follows: “the probability of an event A under the condition of an event B”.
Let's see an example:a friend gives you some device, let it be a telephone. It can be broken (20%) or healthy (80%). Any device that came into your hands can be repaired with a probability of 0.4 or unable 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 is easy to see how it appears in this case.conditional probability: you will not be able to call a person if the phone is broken, and if it is working, you do not need to repair it. Thus, in order to get any results at the “second level”, you need to know which event was executed at the first.
Consider the examples of solving problems on addition and multiplication of probabilities, using the data from the previous paragraph.
First, let's find the probability that youfix the machine given to you. To do this, firstly, it must be faulty, and secondly, you must cope with the repair. This is a typical task using multiplication: we get 0.2 * 0.4 = 0.08.
Finally, consider this option:you got a broken phone, fixed it, then dialed the number, and the person at the opposite end picked up the phone. Here the multiplication of three components is already required: 0.2 * 0.4 * 0.7 = 0.056.
What to do if you have two non-working at oncephone? How likely are you to repair at least one of them? This is a task for addition and multiplication of probabilities, since joint events are used. The solution: 0.4 + 0.4 - 0.4 * 0.4 = 0.8 - 0.16 = 0.64. Thus, if you get two broken machines in your hands, you can handle the repair in 64% of cases.
As mentioned at the beginning of the article, the use of the theory of probability should be deliberate and conscious.
The more a series of experiments, the closerThe theoretically predicted value fits the practice. For example, we throw a coin. Theoretically, knowing the existence of formulas for addition and multiplication of probabilities, we can predict how many times an “eagle” and “tails” fall out if we conduct an experiment 10 times. We conducted an experiment, and by coincidence, the dropout ratio was 3 to 7. But if we conduct a series of 100, 1000 and more attempts, it turns out that the distribution schedule is getting closer to the theoretical: 44 to 56, 482 to 518 and so on.
So, if you turn toto the unknown, to the unexplored area, the theory of probability may not be applicable. Each subsequent attempt in this case may be successful and generalizations like “X does not exist” or “X is impossible” will be premature.
So, we have considered two types of addition, multiplicationand conditional probabilities. Upon further study of this area, it is necessary to learn how to distinguish between situations when each specific formula is used. In addition, you need to be aware of whether probabilistic methods are generally applicable when solving your problem.
