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Problem: The processing area consists of five machines of the same type. The probability that the machine is in order is 0,8 . The planned task can be implemented, if no less than 3 machines are in order. Find the probability that the planned task won't be implemented.

15.6.14 Definition and properties of probability

1.02 $

Problem: The distribution density of the random variable \( X \) has the form \[ f(x)=\left\{\begin{array}{rc} 0, & \text { when } x<1 ; \\ \frac{a}{x^{3}}, & \text { when } 1 \leq x \leq 4 ; \\ 0, & \text { when } x>4 \end{array}\right. \] Find: a) the coefficient \( a \); b) the distribution function \( F(x) \); c) the expected value \( E(X) \) and the dispersion \( D(X) \); d) the probability \( P(3

15.6.15 Definition and properties of probability

2.55 $

Problem: There are \( S \) black and \( N-S \) white balls in the box. Let \( T_{1} \) be the extraction number, in case of which a black ball appears for the 1st time (i.e., \( T_{1} \) is the waiting time). Let's show that \( T_{1} \) has a geometric distribution (the ball, taken out, is returned) and calculate the expected value and the dispersion of \( T_{1} \).

15.6.16 Definition and properties of probability

3.82 $

Problem: The workers operate three machines. Let's denote by \( A_{i} \) the events, that during the day the machine with number \( i \) will require attention, \( i=1,2,3 \). Express the following events through \( A_{i} \) : \( A \) - three machines have a defect during the day; \( B \)-at least one machine has a defect during the day; \( C \)-no machine has a defect during the day.

15.6.17 Definition and properties of probability

0 $

Problem: There are 22 elements in the box, among which 8 have a defect. The assembler randomly takes 4 elements. Find the probability that 3 of the elements that have been taken out, have a defect.

15.6.18 Definition and properties of probability

0.76 $

Problem: There are 8 balls in the \( 1^{\text {st }} \) box, 2 of them are white, and in the \( 2^{\text {nd }} \) box there are 12 balls, 7 of which are white. 1 ball has been randomly taken out of the \( 1^{\text {st }} \) box and put into the \( 2^{\text {nd }} \) one. Find the probability that the ball, taken out of the \( 2^{\text {nd }} \) box after that, will turn out to be white.

15.6.19 Definition and properties of probability

1.27 $

Problem: Out of 1000 lamps 250 belong to the \( 1^{\text {st }} \) batch, 350 belong to the \( 2^{\text {nd }} \) one and 400 to the \( 3^{\text {rd }} \) batch. In the \( 1^{\text {st }} \) batch \( 4 \% \) of the lamps have a defect, in the \( 2^{\text {nd }} \) batch \( 5 \% \) have a defect, and in the \( 3^{\text {rd }} 6 \% \) of the lamps have a defect. A lamp is chosen randomly. Find the probability that the chosen lamp has a defect.

15.6.20 Definition and properties of probability

1.27 $

Problem: The probability of producing an item with a defect is equal to \( 0,003.300 \) items have been bought. Find the probability that the number of items with a defect does not exceed 2 .

15.6.21 Definition and properties of probability

1.02 $

Problem: Find the reliability of the circuit with the given probabilities of a failure-free operation of 5 nodes.

15.6.22 Definition and properties of probability

1.27 $

Problem: Calculate the integral of the complex variable: \[ \int_{L} f(z) d z \text {, where } f(z)=z \bar{z} \text {; } \] \( L \) : the contour of the circular segment with the \( \operatorname{arc} z=e^{i t}, 0 \leq t \leq \pi / 2 \).

10.1.31 Integral of a complex variable

1.27 $

Problem: Using the Cauchy theorem for a multiply connected domain and the Cauchy integral formula, calculate the integral: \[ \int_{L} f(z) d z \text {, where } f(z)=\frac{e^{3 z}}{z^{2}+\frac{\pi^{2}}{9}}, \quad L:|z|=2 . \]

10.1.32 Integral of a complex variable

3.06 $

Problem: Using the Cauchy theorem for the multiply connected domain and the Cauchy integral formula for the analytic function and its derivatives, calculate the integral: \[ \int_{L} f(z) d z \] where \( f(z)=\frac{2 z^{2}-z+1}{(z-i)^{2}(z+i)}, \quad L:|z+i|=3 \).

10.1.33 Integral of a complex variable

3.06 $

Problem: Calculate complex integral using residues: \[ \oint_{C} \frac{1-e^{z}}{z\left(z^{2}-4 z+3\right)} d z, \quad \text { where } \quad C:|z|=4 . \]

10.1.34 Integral of a complex variable

2.04 $

Problem: Calculate complex integral using residues: \[ \oint_{C} \frac{\operatorname{ch} z-1}{z^{2}(z-1)^{2}} d z, \quad C:|z|=2 . \]

10.1.35 Integral of a complex variable

2.04 $

Problem: Find the image of domain \( D \) with linear-fractional mapping \( w=f(z) \), where \( D: \operatorname{Re} z>0, \quad f(z)=2 i \cdot \frac{z-1}{z+1} \).

10.5.7 Conformal mappings

5.1 $

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