Definition and properties of probability/Definition and…

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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

3.9 $

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

0.78 $

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

1.3 $

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

1.3 $

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

1.04 $

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

1.3 $

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