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3.4. Full probability formula

Lecture



The consequence of both main theorems - the theorem of addition of probabilities and the theorem of multiplication of probabilities - is the so-called formula of full probability.

Let it be required to determine the probability of some event.   3.4.  Full probability formula which can happen with one of the following events:

  3.4.  Full probability formula ,

forming a complete group of incompatible events. We will call these events hypotheses.

Let us prove that in this case

  3.4.  Full probability formula , (3.4.1)

those. event probability   3.4.  Full probability formula calculated as the sum of the products of the probability of each hypothesis on the probability of an event with this hypothesis.

Formula (3.4.1) is called the formula of total probability.

Evidence. Since the hypotheses   3.4.  Full probability formula form a complete group then event   3.4.  Full probability formula can only appear in combination with any of these hypotheses:

  3.4.  Full probability formula .

Since the hypotheses   3.4.  Full probability formula incompatible, then combinations   3.4.  Full probability formula also inconsistent; applying to them the addition theorem, we get:

  3.4.  Full probability formula .

Applying to event   3.4.  Full probability formula multiplication theorem, we get:

  3.4.  Full probability formula ,

Q.E.D.

Example 1. There are three identical-looking bins; in the first urn there are two white and one black ball; in the second - three white and one black; in the third - two white and two black balls. Someone chooses one of the urns at random and takes the ball out of it. Find the probability that this ball is white.

Decision. Consider three hypotheses:

  3.4.  Full probability formula - the choice of the first urn,

  3.4.  Full probability formula - the choice of the second urn,

  3.4.  Full probability formula - choice of the third urn

and event   3.4.  Full probability formula - the appearance of a white ball.

Since the hypotheses, according to the condition of the problem, are equally possible,

  3.4.  Full probability formula .

Conditional probabilities of events   3.4.  Full probability formula with these hypotheses are respectively equal:

  3.4.  Full probability formula .

By the formula of total probability

  3.4.  Full probability formula .

Example 2. The plane is three single shots. The probability of hitting the first shot is 0.4, at the second - 0.5, at the third 0.7. To bring the aircraft out of action, three hits are obviously enough; with one hit the plane crashes with a probability of 0.2, with two hits - with a probability of 0.6. Find the probability that as a result of three shots the plane will be disabled.

Decision. Consider four hypotheses:

  3.4.  Full probability formula - not a single shell hit the plane,

  3.4.  Full probability formula - one projectile hit the plane,

  3.4.  Full probability formula - two shells hit the plane,

  3.4.  Full probability formula - Three projectiles hit the plane.

Using the theorems of addition and multiplication, we find the probabilities of these hypotheses:

  3.4.  Full probability formula

Conditional probabilities of events   3.4.  Full probability formula (aircraft failure) with these hypotheses are:

  3.4.  Full probability formula .

Using the formula of total probability, we get:

  3.4.  Full probability formula

Note that the first hypothesis   3.4.  Full probability formula it would be possible not to introduce it into consideration, since the corresponding term in the formula of the full probability vanishes. This is usually the case when applying the formula of full probability, considering not the full group of incompatible hypotheses, but only those of which this event is possible.

Example 3. The engine is controlled by two regulators. Considered a specific period of time   3.4.  Full probability formula during which it is desirable to ensure trouble-free operation of the engine. In the presence of both regulators, the engine refuses with probability   3.4.  Full probability formula , when only the first of them works - with probability   3.4.  Full probability formula , at work only the second -   3.4.  Full probability formula , in case of failure of both regulators - with probability   3.4.  Full probability formula . The first regulator has reliability   3.4.  Full probability formula second -   3.4.  Full probability formula . All elements fail independently of each other. Find the full reliability (probability of failure-free operation) of the engine.

Decision. Consider the hypotheses:

  3.4.  Full probability formula - both regulators work,

  3.4.  Full probability formula - only the first regulator is working (the second one has failed),

  3.4.  Full probability formula - only the second regulator works (the first one is out of order),

  3.4.  Full probability formula - both regulators have failed

and event

  3.4.  Full probability formula - engine uptime.

The probabilities of hypotheses are:

  3.4.  Full probability formula

Conditional probabilities of events   3.4.  Full probability formula with these hypotheses are given:

  3.4.  Full probability formula

According to the formula of the total probability we get:

  3.4.  Full probability formula


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Probability theory. Mathematical Statistics and Stochastic Analysis

Terms: Probability theory. Mathematical Statistics and Stochastic Analysis