Definition and properties of probability

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Problem: Three passengers get on the train, randomly choosing one of the six carriages. What is the probability that at least one of them will get on the first carriage, if it's known that they have got into different carriages?

15.6.1 Definition and properties of probability

1.03 $

Problem: It's known that a 5-digit telephone number has different digits. Under this condition what is the probability that exactly one of the numbers is even ( 0 is considered an even number and the telephone number can begin with a 0 ).

15.6.2 Definition and properties of probability

1.03 $

Problem: The central accounting department of the corporation received packs of receipts for verification and processing. \( 90 \% \) of the packs were found to be satisfactory: they contained \( 1 \% \) of incorrectly formalized receipts. The rest \( 10 \% \) of the receipts were found to be dissatisfying, i.e., they contained \( 5 \% \) incorrectly formalized receipts. What is the probability that a randomly chosen receipt will be incorrectly formalized?

15.6.3 Definition and properties of probability

1.03 $

Problem: There are 5 black and 6 white balls in the box. 4 balls are taken out. Find the probability that there are at least two white balls among those, taken out.

15.6.4 Definition and properties of probability

1.28 $

Problem: The auditor checks three accounts. The probability of correct registration of the account is equal to 0,9 . Find the probability of the events: \( A=\{3 \) accounts are correctly registered \( \} \), \( B=\{2 \) accounts are correctly registered \( \} \), \( C=\{1 \) account is correctly registered \( \} \), \( D=\{ \) at least 1 account is correctly registered \( \} \).

15.6.9 Definition and properties of probability

0.77 $

Problem: A circle with a radius of \( 1 \mathrm{~cm} \) is randomly thrown on the plane, ruled by parallel lines, at the distance of 6 from each other. Find the probability that the circle doesn't cross any of the lines.

15.6.5 Definition and properties of probability

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Problem: \( 30 \% \) of the patients at the hospital have disease \( K \), \( 20 \% \) of the patients have disease \( L \), and the rest of the patients have disease \( M \). The probabilities that the patients with diseases \( K, L, M \) will be discharged healthy, are equal to respectively 0,5 ; 0,\( 4 ; 0,9 \). Find the probability that a healthy patient who was discharged had disease \( K \).

15.6.6 Definition and properties of probability

1.03 $

Problem: There are black, white and blue balls in three boxes. It's known that in the \( 1^{\text {st }} \) box there are 2 white, 4 blue and 5 black balls, in the \( 2^{\text {nd }} \) box there are 4 white, 3 blue and 5 black balls, and in the \( 3^{\text {rd }} \) box there are 1 black, 2 white and 2 blue balls. From a randomly chosen box 3 balls are taken out. Find the probability that they are of the same colour.

15.6.7 Definition and properties of probability

1.28 $

Problem: There are white and black balls in two boxes. There are 2 black and 1 white balls in the \( 1^{\text {st }} \) box, and 3 white and 3 black balls in the \( 2^{\text {nd }} \) box. Two balls are randomly chosen from each of the two boxes and put them in the \( 3^{\text {rd }} \) empty box. It turns out that there are balls of both colours in the \( 3^{\text {rd }} \) box. Find the probability that a ball, randomly chosen from the \( 3^{\text {rd }} \) box, will be white.

15.6.8 Definition and properties of probability

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Problem: There are three batches with the same number of elements. One batch contains second-rate elements, and the other two contain only first-rate elements. A randomly chosen element turned out to be first-rate. Find the probability that it is from the batch, containing second-rate elements.

15.6.10 Definition and properties of probability

2.06 $

Problem: An exam is conducted in the form of a test, consisting of ten questions, each of which has four possible answers, among them exactly one is correct. To pass the test "successfully", at least 3 questions must be given correct answers. In case of failure of the first attempt of passing the test, one retake is allowed in the same form. Find the probability of passing the exam (immediately or after a retake), if the examinee chooses all the answers randomly. Multiply the found value by 1000 , round up to the nearest integer and write the obtained value as an answer.

15.6.11 Definition and properties of probability

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Problem: The deck of 36 cards is divided into two parts of 18 cards. In each part there turned to be two queens. Two cards were transferred from the \( 1^{\text {st }} \) to the \( 2^{\text {nd }} \) parts. Then the young rabbit took a card from the \( 2^{\text {nd }} \) half. Find the probability that it has taken a queen.

15.6.12 Definition and properties of probability

1.28 $

Problem: There are three identical boxes. There are 2 white and 2 black balls in the \( 1^{\text {st }} \) box; there are 3 black balls in the \( 2^{\text {nd }} \) box and 1 black and 5 white balls in the \( 3^{\text {rd }} \) box. Someone, randomly choosing a box, takes out a random ball. What is the probability that the ball will be white?

15.6.13 Definition and properties of probability

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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

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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

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