Poisson formula and an example of solving a problem

Bernoulli's formula is convenient for calculations with only a relatively small number of tests. . For large values use the formula uncomfortable. Most often, in these cases, the Poisson formula is used. By the way, this formula is determined by the Poisson theorem.

Note that the theorem. If the probability event occurrence in each test is constant and small, and the number of independent tests large enough, the probability of an event smooth times approximately equal

, (3.4)

Where .

Evidence. Let given the probability of occurrence in one test and the number of independent trials . Denote . Where p = λ / n . Substitute ϶ᴛᴏ expression in the Bernoulli formula:

For a sufficiently large !! n ,, and relatively small !! m ,, all brackets, with the exception of the penultimate one, can be taken equal to one, i.e.

Given that n is sufficiently large, the right side of this expression can be considered when n-> oo, i.e. find limit

Then we get

(3.5)

Example. The company manufactured and sent to the customer 100,000 bottles of beer. The likelihood that a bottle might be a bat is 0.0001. Find the probability that there will be exactly three and exactly five broken bottles in the shipment.

Decision. Given: n = 100000, p = 0.0001, m = 3 (m = 5).

Find = 10

We use the Poisson formula