1. Probability, Statistics
Practice-1
Additional chapters of mathematics
Dmitriy Sergeevich Nikitin
Assistant
Tomsk Polytechnic University

2. Rating plan for the discipline "Additional chapters of mathematics"№
Type of classes
Estimated parameters
Quantity
Unit cost
Total score 1.
Lectures
Visiting and lecture notes 12
0,5 6
Theoretical tests 4 5 20 2.
Practical lessons
Visiting and working 10
Control works 24
Individual homework 4.
Exam
1) Theoretical question.
2) Task.
3) Task.
(Extra points) - Report
(10)
TOTAL
100 (+10)

3. Random variable
In an experiment we observe a random variable X, that is, a function whose values in a trial (a performance of an experiment) occur “by chance” according to a probability distribution that gives the individual probabilities with which possible values of X may occur in the long run. 3

4. Random variable Examples of random variables:
the number of failures of the power supply system over a period of time;
the time of finding the damage and repairing the failed cable.
Random variables that take only values separated from each other that can be enumerated in advance are called discrete random variables (the first example). Random variables whose possible values continuously fill a certain interval are called continuous random variables (the second example). 4

5. Actions with events and variables
The sum of two events A and B is the event C, consisting of the performance of event A or event B or both. In other words, the sum of two events A and B is the event C, consisting in the appearance of at least one of the events A or B. The sum of several events is an event consisting in the appearance of at least one of these events.
The product of two events A and B is the event C, consisting in the joint execution of event A and event B. The product of several events is the event consisting in the joint appearance of all these events. 5

6. Actions with events and variables

7. Actions with events and variables
The probability of the sum of two disjoint Events events is equal to the sum of the probabilities of these events: 7

8. Actions with events and variables
When events A and B are joint, the probability of the sum of these events is expressed by the formulas: 8

9. Actions with events and variables
When events A and B are joint, the probability of the product of these events is expressed by the formulas: 9

10. Task 1
The figure shows the power supply system consisting of a power supply G, a step-up transformer T1, a power line L and a step-down transformer T2.
The probability of failure of the elements of the power supply system is denoted by q with the corresponding index.
Define in general the probability of failure of the power supply system.

11. Task 2
The device consists of three elements including two elements of the first type A1 and A2 and one element of type B. Elements A1 and A2 duplicate each other: if one of them fails, the automatic switch to the second one. Element B is not duplicated. The device refuses, if both elements A1 and A2 or element B are simultaneously refuse.
It is required to express the probability of the event C (device failure) in terms of the probabilities of events containing only sums and not products of elementary events A1 and A2 or device B.

12. Conditional Probability
Often it is required to find the probability of an event B under the condition that an event A occurs. This probability is called the conditional probability of B given A and is denoted by P(B|A). In this case A serves as a new (reduced) sample space, and that probability is the fraction P(A) of which corresponds to 𝐀∩𝐁. Thus
P(B|A)= 𝐏 𝐀𝐁 𝐏 𝐀 .
Similarly, the conditional probability of A given B is
P(A|B)= 𝐏 𝐀𝐁 𝐏 𝐁 .
T H E O R E M 4. Multiplication Rule. If A and B are events in a sample space S and 𝐏 𝐀 ≠𝟎, 𝐏 𝐁 ≠𝟎, then
𝐏 𝐀𝐁 =𝐏 𝐀 𝐏 𝐁|𝐀 =𝐏 𝐁 𝐏 𝐀|𝐁 .

13. Independent Events Independent Events. If events B and A are such that
𝐏 𝐀𝐁 =𝐏 𝐀 𝐏 𝐁
they are called independent events. Assuming 𝐏 𝐀 ≠𝟎, 𝐏 𝐁 ≠𝟎,
P(A|B)=𝐏 𝐀 , P(B|A)=𝐏 𝐁 .
This means that the probability of A does not depend on the occurrence or nonoccurrence of B, and conversely. This justifies the term “independent.”
Independence of m Events. Similarly, m events 𝐀 𝟏 ,…, 𝐀 𝐦 are called independent if
𝐏 𝐀 𝟏 … 𝐀 𝐦 =𝐏 𝐀 𝟏 …𝐏 𝐀 𝐦 .

14. Probability of product
The probability of the product of several events is equal to the product of the probabilities of these events, and the probability of each next event in the order is calculated under the condition that all the previous events took place:
The probability of the product of independent events is equal to the product of the probabilities of these events

15. Task 3
The power distribution unit have four outgoing lines to consumers. Consumers have a nominal load: P kW; P kW; P kW; P kW. The probability of the included state of the consumers is, respectively, P1 = 0.3; Р2 = 0,4;
P3 = 0.2; P4 = 0.8.
Determine the probability that the power supply cable will be loaded at 100 kW.

16. Task 4
Excavator produces a cable trench opening. At the same time it hooked the cable three times. Probability of cable damage at the first engagement P1 = 0,4; at the second - Р2 = 0,5; at the third - Р3 = 0,7.
Find the probability that as a result of these three links the cable will be damaged:
a) exactly once;
b) at least once.

17. End of Practice-1