Lecture 13 Bernoulli’s Formula. The Most Probable Number of Occurrences of an Event If n

The difference between the end point values in this double inequality is 1:

.

If is not an integer, then the double inequality specifies only one most probable value . Now if is an integer, then there are two most probable values,

and

Theorem. The probability that the event will occur (in n independent trials):

1. 
   less than k times ( ) is found by the formula
   

2. 
   more than k times ( ) is found by the formula
   

3. 
   no less than k times ( ) is found by the formula
   

4. 
   no more than k times ( )is found by the formula
   

.