Bernoulli formula and an example problem solution

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 ( ).

Let, in general, be produced independent trials. The task is to determine the probability that exactly trials event will come if the probability of occurrence of this event in each trial is equal . 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 trials event come, and in the rest 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.

,

Where .

Since only one of the possible combinations was considered when an event happened only in the first 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 items by i.e. .

So the probability that an event come exactly in tests determined by the formula

, (3.3)

Where .

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