Practical lesson 2. Bernoulli's formula
Practical session "Repeated independent experiments. Bernoulli's formula. "
Remember that Bernoulli’s scheme is repetition of the same experiment independently.
At the same time, in each test we study event A, which probability is p, if event A happens, we say that we had a success, and if event A does not occur, there was a failure.
Most often Bernoulli scheme is considered either until the first success, or for a fixed number of experiment repetitions we examine the probability of occurring the given number of successes.
Consider these examples.
The first example.
The shooter fires at a target till the first hit.
The probability that one shot hits the target is 0.7.
What is the probability that he would have to shoot 4 times?
Let Ai is a success in the i-experiment.
Then event B - "the shooter shot 4 times", can be viewed as a failure for the first time, a failure for the second time, a failure for the third time and the successful fourth time.
Since all of these events are independent, then the probability of event B is the product of probabilities.
The first failure, the second failure, the third failure, the fourth success.
We also agreed to denote by q = 1-p, then there is a chance of failure.
Then the probability of our events will be in the p * q3.
In our case r = 0.7, then q = 0,3. The probability will be equal to 0.7 * 0.33.
The result can be counted with a calculator.
The next task.
There we already know that the shooter does 4 shots at the target, thus again assume that the shots are independent with respect to each other.
The probability of hitting a target with one shot is 0.8.
We need to find the probability that the shooter misses twice.
What is a "to miss twice"?
The same thing as hit twice.
Then you can use the Bernoulli formula, that is, if we have p = 0.8, i.e. success is hitting the target, then q = 0,2.
Then we need to find the probability that, with a probability of 0.8, 2 shots of 4 will hit the target (see video).
You may find the result yourselves.
Another variant.
The shooter will miss at least twice, it means that there will be 2, 3 or 4 misses.
It can be calculated fairly.
WThat is what is: 2, 3 or 4 misses?
This means that the number of successes is 2, 1 or 0.
So we need to find the following sum: the probability that there are two hits of four shots, plus the probability that there is one hit with three misses of four shots and, plus the probability that there are 0 hits of four shots.
We write down the probability by index, sometimes it is dropped.
Again we use the Bernoulli formula (see video).
It can all be calculated with a calculator and we get the answer.
And if there were not 4 shots, but, for example, 10, then the probability that the shooter will miss at least twice should be counted that he will miss 2, 3, 4, 5, ..., 10.
A lot of calculations.
Sometimes it is more convenient not to calculate the probability of the event itself, but the probability of its complement, that is, to consider the probability that the shooter will miss at least twice, that is zero or one - this means that he will hit 3 or 4 times, and subtract the obtained value from 1 .
We write down: 1- (R0,4 (4,3) + R0,8 (4,4)).
In this example, we will not calculate the result, do it yourselves.
Check that both results are identical.
We checked at the lecture that the sum of all probabilities according to Bernoulli’s formula is when k between 0 and n, is equal to 1.
Therefore, this formula is true.
The next problem.
A student is late for the lecture with a probability of 0.9.
There are 25 students in the course.
It is required to find the most probable number of students at the lecture.
The most probable number is the number of students, who will be at the lecture with maximum probabilty.
That is, we wrote down the probability that 0 students out of n will come, the probability that 1 student out of n will come to the lecture ..., the probability that all students of the n will come.
From these numbers, we need to find the largest k.
Note here that these probabilities behave as follows: first, they increase and then decrease, so we have either one or two such numbers.
At the lecture, we showed that tis most probable number k0 lies within the boundaries (np-q) and (np + q).
That is, we do not need to calculate all these probabilities and search the maximum among them.
It is sufficient only to assess the border.
In the length interval 1(p + q = 1) there can be either one or two integers, k0 is necessarily an integer.
In our case it is an integer.
In our case, n = 25, p is the probability that a student will come, this probability is 0.1, minus the probability that he will be late is 0.9.
Then we get 25 * 0.1-0.9 = 1.6.
Then in the interval [1.6; 2,6] there is only one integer, this is 2.
Hence, the most probable number of students who will come to the beginning of the lecture is 2.
Here, of course, the probability is so bold 0.9.
This formula can also be used, as we have proved it.
You are ready for solving problems.