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TASK LIST NO. 1: Random Events and Probability

Task 1

Let the sample space \(\Omega\) of elementary events of an experiment consist of five elementary events \(\omega_i\): \(\Omega=\{\omega_1, \omega_2, \omega_3, \omega_4, \omega_5\}\). We define the events: \(A=\{\omega_1, \omega_3, \omega_5\}\), \(B=\{\omega_2, \omega_3, \omega_4\}\).

Find the events:

  1. \(A \cup B\) (union of events)
  2. \(A \cap B\) (intersection of events)
  3. \(B \backslash A\) (difference of events)
  4. \(A \backslash B\)

Task 2

Consider an electrical circuit in which element \(a_1\) is connected in series with a block consisting of two elements \(a_2\) and \(a_3\) connected in parallel. Let \(A_i, i=1, 2, 3\), denote the event "element \(a_i\) is functional at time \(t\)".

Using operations on events \(A_i\) and the symbols for union (\(\cup\)) and intersection (\(\cap\)), describe the event \(A\): "in the time interval \(t\), the current flow through the circuit will not be interrupted".

Task 3

Person \(X\) performs a certain job in 4, 5, or 6 hours and may commit 0, 1, or 2 errors. Assuming equal probability for each of the 9 possible elementary events (pairs: time, number of errors), find the probability of the following events:

  1. The job will be completed in 4 hours.
  2. The job will be completed flawlessly in 6 hours.
  3. The job will be completed in at most 5 hours.
  4. The job will be completed in at most 5 hours and with at most one error.

Task 4

A fragment of an electrical network consists of two elements connected in parallel: \(a_1\) and \(a_2\). Let \(A_i, i=1, 2\), denote the event that element \(a_i\) remains functional for at least time \(t\).

Calculate the probability of continuous current flow through this system for at least time \(t\), given that \(P(A_1)=P(A_2)=p\) and the probability of simultaneous functionality of both elements is \(P(A_1 \cap A_2)=p^2\).

Task 5

We consider the volume (in \(dm^3\)) of water that a concrete culvert can conduct per second. Past observations allow us to assume that:

  • The probability that the volume of water takes a value from the interval \(\langle 125, 250 \rangle\) is \(P(A) = 0.6\).
  • The probability that the volume of water takes a value from the interval \((200, 300\rangle\) is \(P(B) = 0.7\).
  • The probability of the union of these events is \(P(A \cup B)=0.8\).

Calculate the probability:

  1. \(P(A')\) (complementary event to A)
  2. \(P(A \cap B)\) (intersection of intervals)
  3. \(P(A' \cap B')\) (water volume does not fall into either of these intervals)

Task 6

Identical products manufactured by 2 automated machines are placed on a conveyor belt. The quantitative ratio of production of the first machine to the production of the second is \(3:2\). The first machine produces on average \(65\%\) of first-grade products, while the second produces \(85\%\).

  1. One product is randomly selected from the products on the conveyor belt. Calculate the probability that it will be a first-grade product (use the total probability formula).
  2. A randomly selected product turned out to be of first quality. Calculate the probability that it was produced by the first machine (use Bayes' theorem).

Task 7

On a communication line, two types of signals are transmitted in the form of code combinations 111 or 000 with a priori probabilities of \(0.65\) and \(0.35\) respectively. The signals are subject to random interference, as a result of which the symbol 1 can be received as 0 with a probability of \(0.2\), and with the same probability, the symbol 0 can be received as 1. We assume that symbols 1 and 0 are subject to interference independently of each other.

Calculate the probability of receiving the signal:

  1. 111
  2. 000
  3. 010

Task 8

Coded information consists of seven pulses of types \(A, B, C\) in quantities: four pulses of \(A\), two pulses of \(B\), and one pulse of \(C\). Assuming a random arrangement of pulses, find the probability that:

  1. the first received pulse will be \(A\),
  2. the first received pulse will be \(A\) or \(C\),
  3. the first two pulses will be \(A\) and \(C\) in that order.

Task 9

A certain good is produced by 3 plants. The probability of producing first-quality goods by these plants is \(0.97\), \(0.90\), and \(0.86\), respectively.

Find the probability that a randomly taken item — from among three items originating (one each) from different plants — is of first quality.

Task 10

Only 3 types of letter sequences are transmitted via a communication channel: \(AAAA\), \(BBBB\), \(CCCC\) with probabilities \(0.4\), \(0.3\), and \(0.3\), respectively. These letters (signals) are subject to independent random interference (errors), as a result of which, e.g., letter \(A\) can be received as \(B\) or \(C\). The probabilities of correct transmission or error for a single letter are given in the table:

Transmitted \ Received A B C
A 0.8 0.1 0.1
B 0.1 0.8 0.1
C 0.1 0.1 0.8

Find the probability of receiving the signal at the output:

  1. \(AAAA\)
  2. \(ACAA\)