Glossary

opposite events – противоположные события; to extract – извлекать

without replacement – без возвращения; conditional probability – условная вероятность

preceding – предшествующий; conic – конический; elliptic – эллиптический

cylinder – валик; collector – сборщик

Exercises for Seminar 3

3.1. In a cash–prize lottery 150 prizes and 50 monetary winnings are played on every 10000 tickets. What is the probability of a winning indifferently monetary or prize for an owner of one lottery ticket equal to?

3.2. The events A, B, C and D form a complete group. The probabilities of the events are those: P(A) = 0,1; P(B) = 0,4; P(C) = 0,3. What is the probability of the event D equal to?

3.3. The probability that a shooter will beat out 10 aces at one shot is equal to 0,1 and the probability to beat out 9 aces is equal to 0,3. Find the probabilities of the following events: A – the shooter will beat out 8 or less aces; B – the shooter will beat out no less than 9 aces.

3.4. There are 10 details in a box, and 2 of them are non-standard. Find the probability that in randomly selected 6 details appears no more than one non-standard detail.

Direction: If A is «there is no non-standard details» and B is «there is one non-standard detail» then P(A + B) = P(A) + P(B) =…

3.5. An enterprise produces 95% standard products, and 86% of them have the first grade. Find the probability that a randomly taken product made at the enterprise will be standard and the first grade (grade – сорт).

3.6. Two dice are thrown. Find the conditional probability that each die lands on 5 if it is known that the sum of aces is divided on 5.

3.7. If two dice are rolled, what is the conditional probability that the first one lands on 4 given that the sum of the dice is 8?

3.8. In a certain community, 36 percent of the families own a dog, and 22 percent of the families that own a dog also own a cat. In addition, 30 percent of the families own a cat. What is

(a) the probability that a randomly selected family owns both a dog and a cat;

(b) the conditional probability that a randomly selected family owns a dog given that it owns a cat?

3.9. An ordinary deck of 52 playing cards is randomly divided into 4 piles of 13 cards each. Compute the probability that each pile has exactly 1 ace (a pile – стопка; a deck – колода; an ace – туз).

3.10. A coin is tossed until it will not land on the same side 2 times in succession. Find the probability that the experiment will terminate before the sixth tossing (in succession – подряд).

Exercises for Homework 3

3.11. By the statistical data of a repair shop 20 stops of a lathe are on the average: 10 – for change of a cutter; 3 – because of malfunction of a drive; 2 – because of delayed submission of details. The rest stops occur for other reasons. Find the probability of stop of the lathe for other reasons (repair shop – ремонтная мастерская; lathe – токарный станок; cutter – резец; malfunction – неисправность; drive – привод). The answer: 0,25.

3.12. There are 30 TVs in a shop, and 20 of them are import. Find the probability that no less than 3 import TVs will be among 5 TVs sold for one day, assuming that the probabilities of purchase of TVs of different marks are identical. The answer: 0,81.

3.13. If two dice are rolled, what is the conditional probability that the first one lands on 6 given that the sum of the dice is 11? The answer: 0,5.

3.14. An urn contains 6 white and 9 black balls. If 4 balls are to be randomly selected without replacement, what is the probability that the first 2 selected are white and the last 2 black?

The answer: 0,066.

3.15. Fifty-two percent of the students at a certain university are females. Five percent of the students in this university are majoring in computer science. Two percent of the students are women majoring in computer science. If a student is selected at random, find the conditional probability that

(a) this student is female, given that the student is majoring in computer science;

(b) this student is majoring in computer science, given that the student is female.

The answer: a) 0,4; b) 0,038.

3.16. Celine is undecided as to whether to take a French course or a chemistry course. She estimates that her probability of receiving an A grade would be 1/2 in a French course and 2/3 in a chemistry course. If Celine decides to base her decision on the flip of a coin, what is the probability that she gets an A in chemistry (grade – оценка)? The answer: 0,33.

3.17. Suppose that an urn contains 8 red and 4 white balls. We draw 2 balls from the urn without replacement. If we assume that at each draw each ball in the urn is equally likely to be chosen, what is the probability that both balls drawn are red (to draw – тянуть)? The answer: 0,424.

3.18. An urn contains 10 white, 15 black, 20 blue and 25 red balls. A ball is taken at random from the urn. Find the probability that the taken ball is: a) white or black; b) blue or red.

3.19. Two shooters shoot in a target. Find the probability that the target will be struck at least one of the shooters if it is known that the probability of miss by both shooters is equal to 0,27.

3.20. Two cards are randomly selected from a pack of 36 playing cards. Find the probability that both cards are the same color (a pack – колода). The answer: 0,486.