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Bernoulli formula and an example problem solution

Lecture



Suppose that several identical machines in the same conditions transport cargo. Any car can fail during these shipments. Let the probability of failure of one machine does not depend on the failure of other machines. This means that independent events (trials) are considered. The probability of failure of each of these machines will take the same (   Bernoulli formula and an example problem solution ).

Let, in general, be produced   Bernoulli formula and an example problem solution independent trials. The task is to determine the probability that exactly   Bernoulli formula and an example problem solution trials event will come   Bernoulli formula and an example problem solution if the probability of occurrence of this event in each trial is equal   Bernoulli formula and an example problem solution . In the case of cars, this may be the probability of failure of exactly one car, exactly two cars, etc.

We first define the probability that in the first   Bernoulli formula and an example problem solution trials event   Bernoulli formula and an example problem solution come, and in the rest   Bernoulli formula and an example problem solution trials - will not come. The probability of such an event can be obtained on the basis of the formula for the probability of the production of independent events.

  Bernoulli formula and an example problem solution ,

Where   Bernoulli formula and an example problem solution .

Since only one of the possible combinations was considered when an event   Bernoulli formula and an example problem solution happened only in the first   Bernoulli formula and an example problem solution tests, then to determine the desired probability you need to go through all possible combinations. Their number will be equal to the number of combinations of   Bernoulli formula and an example problem solution items by   Bernoulli formula and an example problem solution i.e.   Bernoulli formula and an example problem solution .

So the probability that an event   Bernoulli formula and an example problem solution come exactly in   Bernoulli formula and an example problem solution tests determined by the formula

  Bernoulli formula and an example problem solution , (3.3)

Where   Bernoulli formula and an example problem solution .

Formula (3.3) is called the Bernoulli formula.

Example. In four attempts, some items are played. The probability of winning each attempt is known and is 0.5. What is the probability of winning exactly three items?

Decision. According to the formula Bernoulli find

  Bernoulli formula and an example problem solution


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Probability theory. Mathematical Statistics and Stochastic Analysis

Terms: Probability theory. Mathematical Statistics and Stochastic Analysis