Operations over events
We enter operations over events: sum, product and negation.
The sum of two events A and B is such third event A + B which consists in appearance of at least one of these events, i.e. A or B. If A and B are compatible events then their sum A + B means appearance of either the event A, or the event B, or both A and B. If A and B are incompatible events then their sum A + B means appearance of either the event A, or the event B. For example, if two shots are made by a gun and A is hit at the first shot, B is hit at the second shot then A + B is either hit at the first shot or hit at the second shot, or two hits. The event A + B + C consists of appearance of one of the following events: A; B; C; both A and B; both A and C; both B and C; all the events A, B and C.
The product of two events A and B is such third event AB which consists in simultaneous appearance of the events A and B. If A and B are incompatible events then their product AB is an impossible event.
The
negation
of an event A
is the event
(not A)
which consists in non-appearance of the event A.
Observe that A
+
is a reliable event, and A
is an impossible event.
Example. A winner of a competition is rewarded: by a prize (the event А), a money premium (the event В), a medal (the event С).
What
do the following events represent: a) A
+ B;
b) ABC;
c)
?
Solution: a) The event A + B consists in rewarding the winner by a prize, or a money premium, or simultaneously both a prize and a money premium.
b) The event ABC consists in rewarding the winner by a prize, a money premium and a medal simultaneously.
c) The event consists in rewarding the winner by both a prize and a medal simultaneously without giving a money premium.
Glossary
seemingly – на вид, по-видимому; arrangement – расположение
allocation – размещение; combination – сочетание
permutation – перестановка; three-place number – трехзначное число
consecution – следование; repetition – повторение
to extract – извлекать; simultaneous – одновременный
negation – отрицание
Exercises for Seminar 2
2.1. A college planning committee consists of 3 freshmen, 4 sophomores, 5 juniors, and 2 seniors. A subcommittee of 4, consisting of 1 person from each class, is to be chosen (a freshman – первокурсник; a sophomore – второкурсник). How many different subcommittees are possible?
2.2. How many outcome sequences are possible when a die is rolled four times, where we say, for instance, that the outcome is 3, 4, 3, 1 if the first roll landed on 3, the second on 4, the third on 3, and the fourth on 1?
2.3. There are five disks on the general axis of a lock. Each disk is subdivided into six sectors on which different letters are written. The lock opens only in the event that each disk occupies one certain position regarding the case of the lock. Find the probability that the lock can be opened at any installation of disks (the case of a lock – корпус замка).
The answer: 0,0001286.
2.4. The order of performance of 7 participants of a competition is determined by a toss-up. How many different variants of the toss-up are possible?
2.5. There are 10 cards each of which contains one letter: 3 cards with letter A, 2 cards with letter S and 5 cards with letters D, R, O, M, B. A child takes cards in a random order and puts one to another. Find the probability that the word «AMBASSADOR» will be turned out (to turn out – оказываться).
2.6. By the conditions of the lottery «Sportloto 6 of 45» a participant of the lottery who have guessed 4, 5 or 6 numbers from 6 randomly selected numbers of 45 receives a monetary prize. Find the probability that the participant will guess: a) all 6 numbers; b) 4 numbers.
2.7. 10 of 30 students have sport categories. What is the probability that 3 randomly chosen students have sport categories?
The answer: 0,03.
2.8. A group consists of 12 students, and 8 of them are pupils with honor. 9 students are randomly selected. Find the probability that 5 pupils with honor will be among the selected.
The answer: 0,255.
2.9. Eight different books are randomly placed on one shelf. Find the probability that two certain books will be put beside (a shelf – полка, beside – рядом).
The answer: 0,25.
2.10. A box contains 5 red, 3 green and 2 blue pencils. 3 pencils are randomly extracted from the box. Find the probabilities of the following events:
A – all the extracted pencils are different color;
B – all the extracted pencils are the same color;
C – one blue pencil among the extracted;
D – exactly two pencils of the same color among the extracted.
The answer: P(A) = 0,25; P(B) = 0,092; P(C) = 0,467; P(D) = 0,658.
2.11. It has been sold 21 of 25 refrigerators of three marks available in quantities of 5, 7 and 13 units in a shop. Assuming that the probability to be sold for a refrigerator of each mark is the same, find the probability of the following events:
a) refrigerators of one mark have been unsold;
b) refrigerators of three different marks have been unsold.
The answer: a) 0,06; b) 0,396.
2.12. A shooter has made three shots in a target. Let Ai be the event «hit by the shooter at the i-th shot» (i = 1, 2, 3). Express by A1, A2, A3 and their negations the following events: A – «only one hit»; B – «three misses»; C – «three hits»; D – «at least one miss»; E – «no less than two hits»; F – «no more than one hit».