A consequence of the multiplication theorem and the full probability formula is the so-called hypothesis theorem, or the Bayes formula.
Let’s set the following task.
There is a complete group of incompatible hypotheses. The probabilities of these hypotheses before experience are known and equal, respectively. An experiment was made, as a result of which the occurrence of some event was observed. The question is, how should the probabilities of hypotheses be changed due to the occurrence of this event?
Here, essentially, it is about finding a conditional probability for each hypothesis.
From the multiplication theorem we have:
or by discarding the left side,
Expressing with the help of the formula of the total probability (
Example 1. The device can be assembled from high-quality parts and from parts of normal quality; in general, about 40% of devices are assembled from high-quality parts. If the device is assembled from high-quality parts, its reliability (probability of failure-free operation) during the time is 0.95; if from parts of normal quality, its reliability is 0.7. The device was tested over time and worked flawlessly. Find the probability that it is assembled from high-quality parts.
Decision. Two hypotheses are possible:
– The device is assembled from high quality parts
– The device is assembled from parts of normal quality.
The probability of these hypotheses to experience:
As a result of the experiment, an event was observed – the device worked smoothly for time.
The conditional probabilities of this event under hypotheses and are equal to:
According to the formula (
Example 2. Two shooters independently fire at one target, each making one shot. The probability of hitting the target for the first arrow is 0.8, for the second 0.4. After shooting, one hole was found in the target. Find the probability that this hole belongs to the first arrow.
Decision. Before the experiment, the following hypotheses are possible:
– neither the first nor the second shooter will fall,
– both arrows will hit
– The first shooter will hit, and the second will not,
– The first shooter will not fall, and the second will.
The probability of these hypotheses:
The conditional probabilities of the observed event with these hypotheses are equal:
After the experiment, the hypotheses become impossible, and the probabilities of the hypotheses will be equal:
Consequently, the probability that the hole belongs to the first arrow equals.
Example 3. Some object is monitored using two observation stations. The object can be in two different states and, randomly moving from one to another. By long-term practice it has been established that approximately 30% of the time an object is in a state, and 70% is in a state. Observation station No. 1 transmits erroneous information in approximately 2% of all cases, and observation station No. 2 in 8%. At some point in time, Observation Station No. 1 reported: the object is in a state, and Observation Station No. 2: the object is in a state.
The question is: which of the messages to believe?
Decision. Naturally, to believe the message, for which the greater likelihood that it is true. Let’s apply the Bayes formula. To do this, we make hypotheses about the state of the object:
– the object is in a state
– the object is in a state.
The observed event is as follows: station number 1 reported that the object is in the state, and station number 2 – that it is in the state. Probabilities of hypotheses before experience
Let us find the conditional probabilities of the observed event under these hypotheses. Under the hypothesis that an event should occur, the first station needs to transmit the correct message, and the second one erroneous:
Applying the Bayes formula, we find the probability that the true state of the object is: