Theory of Probability
By
Dr. Brajesh Kumar Jha
Associate Professor
Department of Mathematics
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1.1. Theory of Probability
Introduction: If an experiment is repeated under essentially homogeneous and similar conditions, we generally come across two types of situations:
(a) The result or what is usually known as the ‘outcome’ is unique or certain. The phenomena covered under this trial are known as ‘deterministic’ or ‘predictable’ phenomena. By a deterministic phenomenon we mean one in which the result can be predicted with certainty.
Example:
(i) For a perfect gas, , i.e., PV=constant, where V is the volume and P is the pressure of the gas, provided the temperature remains the same.
(ii) The velocity ‘v’ of a particle after time t is given by where u is the initial velocity and ‘a’ is the acceleration. The equation uniquely determines v if the right hand quantities are known.
(b) The result is not unique but may be one of the several possible outcomes. The phenomena are ‘unpredictable’ or ‘probabilistic’.
(i) In a random toss of a uniform coin we are not sure of getting the head or tail.
(ii) A manufacturer cannot ascertain the future demand of his product with certainty.
(iii) A sales manager cannot predict with certainty about the sales target next year.
(iv) If an electric tube has lasted for one year, nothing can be predicted about its future life.
Probability is also used informally in day to day life. We daily come across the sentences like:
(i) Possibly, it will rain today.
(ii) There is a high chance of my getting the job next month.
(iii) This year’s demand for the product is likely to exceed that of the last year’s.
(iv) The odds are 3:2 in favour of getting the contract applied for.
“Probability is the science of decision making with calculated risks in the face of uncertainty”.
1.2. Basic Terminology
1.1. Random Experiment: If in each trial of an experiment conducted under identical conditions, the outcome is not unique, but may be any one of the possible outcomes, then such an experiment is called a random experiment.
Example: tossing a coin, throwing a die, selecting a card from a pack of playing cards, selecting a family out of a given group of families, etc.
A pack of cards consists of four suits called spades, Hearts, Diamonds and Clubs. Each suits consists of 13 cards, of which nine cards are numbered from 2 to 10, an ace, king, a queen, and a jack. Spades and clubs are black faced cards, while hearts and diamonds are red faced cards.
1.2. Outcome: The result of a random experiment will be called an outcome.
1.3. Trial and Event: Any particular performance of a random experiment is called a trial and outcome or combination of outcomes are termed as events.
Example:
(i) If a coin is tossed repeatedly, the result is not unique. We may get any of the two faces, head or tail. Thus tossing of a coin is a random experiment or trial and getting of a head or tail is event.
(ii) In an experiment which consists of the throw of a six faced die and observing the number of points that appear, the possible outcomes are 1,2,3,4,5,6.
1.4. Exhaustive Events or Cases: The total number of possible outcomes of a random experiment is known as the exhaustive events or cases.
Example:
(i) In tossing of a coin, there are two exhaustive cases, viz., head and tail (the possibility of the coin standing on an edge being ignored).
(ii) In throwing of a die, there are 6 exhaustive cases since any one of the 6 faces 1,2,…6 may come uppermost.
(iii) In drawing two cards from a pack of cards, the exhaustive number of cases is 52C2, since 2 cards can be drawn out of 52 cards in 52C2 ways.
(iv) In throwing of two dice, the exhaustive number of cases is , since any of the numbers 1 to 6 on the first die can be associated with any of the 6 numbers on the other die. In general, in throwing n dice, the exhaustive number of cases is 6n .
1.5. Favourable Events of
Cases: The number of cases favourable to an
event in a trial is the number of outcomes which entail the happening of the
event.
Example:
(i) In drawing a card from a pack of cards the number of cases favourable to drawing of an ace is 4, for drawing a spade is 13 and for drawing a red card is 26.
(ii) In throwing of two dice, the number of cases favourable to getting the sum 5 is:
(1,4),(4,1),(2,3),(3,2), i.e., 4
1.6. Mutually Exclusive Events: Events are said to be mutually exclusive or incompatible if the happening of any one of them precludes the happening of all the others, i.e., if no two or more of them can happen simultaneously in the same trial. For example
(i) In throwing a die all the 6 faces numbered 1 to 6 are mutually exclusive since if any one of these faces comes, the possibility of others, in the same trial, is ruled out.
(ii) Similarly in tossing a coin the events head and tail are mutually exclusive.
1.7. Equally likely Events: Outcomes of trial are said to be equally likely if taking into consideration all the relevant evidences, there is no reason to expect one in preference to the others. For example,
(i) In a random toss of an unbiased or uniform coin, head and tail are equally likely events.
(ii) In throwing an unbiased die, all the six faces are equally likely to come.
1.8. Independent Events: several events are said to be independent if the happening (or non-happening) of an event is not affected by the supplementary knowledge concerning the occurrence of any number of the remaining events. For Example
(i) In tossing an unbiased coin, the event of getting a head in the first toss is independent of getting a head in the second, third and subsequent throws.
(ii) When a die is thrown twice, the result of the first throw is independent of getting a head in the second throw.
(iii) If we draw a card from a pack of well-shuffled cards and replace it before drawing the second card, the result of the second draw is independent of the first draw. But, however, if the first card drawn is not replaced then the second draw is dependent on the first draw.
1.3. Mathematical Probability
Definition: If a random experiment or a trial results in ‘n’ exhaustive, mutually exclusive and equally likely outcomes (or cases), out of which m are favourable to the occurrence of an event E, then the probability ‘p’ of occurrence (or happening) of E, usually denoted by P(E), is given by:
Remark:
(i) Since and so
and
(ii) Sometimes m/n is expressed by saying that ‘the odds in favour of E are m: (n-m) or the odds against E are (n-m) : m’.
(iii) The non-happening of the event E is called the complementary event of E and is denoted by . The number of cases favourable to , i.e., non happening of E is (n-m). then the probability q that E will not happen is given by:
(iv) Probability ‘p’ of the happening of an event is also known as the probability of success and the probability ‘q’ of the non- happening of the event as the probability of failure, i.e., (p+q=1)
(v) If then E is called a certain event and if , E is called an impossible event.
Q1. What is the chance that a leap year selected at random will contain 53 Sundays?
Ans: 2/7
Q2. Two unbiased dice are thrown. Find the probability that:
(i) Both the dice show the same number,
Ans: 1/6.
(ii) The first die shows 6,
Ans: 1/6.
(iii) The total of the numbers on the dice is 8,
Ans: 5/36
(iv) The total of the numbers on the dice is greater than 8,
Ans: 5/18.
(v) The total of the numbers on the dice is 13,
Ans: 0
(vi) The total of the numbers on the dice is any numer from 2 to 12, both inclusive.
Ans: 1.
Q3. Among the digits 1,2,3,4,5 at first one is chosen and then a second selection is made among the remaining four digits. Assuming that all twenty possible outcomes have equal probabilities, find the probability that an odd digit will be selected
(i) The first time
(ii) The second time
(iii) Both times
Ans: (i) 3/5 (ii) 3/5 (iii) 3/10
Q4. From 25 tickets, marked with first 25 numerals, one is drawn at random. Find the chance that
(i) It is multiple of 5 or 7,
Ans: 8/25
(ii) It is multiple of 3 or 7,
Ans: 2/5
1.4. Permutations and Combinations
Permutations: Given n different things (elements or objects), we may arrange them in a row in any order. Each such arrangement is called a permutations of the given things. For example we have 6 permutations of the three letters a, b, c, namely, abc, acb, bac, bca, cab, cba,
Theorem1. The number of permutations of n different things taken all at a time is n!=1.2.3….n
Example 1. If there are 10 different screws in a box that needed in a certain order for assembling a certain product, and these screws are drawn at random from the box, the probability P of picking them in the desired order is very small, namely
When not all given things are different, we obtain for their number of permutations as follows
Theorem2. If n given things can be divided into c classes such that things belonging to the same class are alike while things belonging to different classes are different, then the number of permutations of these things taken all at a time is,
Where nj is the number of things in the jth class.
Example: In how many ways can the letters of REARRANGE be permuted?
Solution: There are nine letters with three R’s, two A’s and two E’s. the number of permutations is.
Example2: If a box contains 6 red and 4 blue balls, the probability of drawing first the red and then the blue balls is.
Theorem 3: The number of different permutations of n different things taken k at a time without repetition is
(3a)
And with repetitions is nk . (3b)
Example: In a coded telegram the letters are arranged in groups of five letters, called words, from (3b) we see that the number of different such word is . From (3a) it follows that the number of different such words containing each letter no more than once is, .
Combinations: In a permutation, the order of the selected things is essential. In contrast, a combination of given things means any selection of one or more things without regard to order.
There are two kinds of combinations:
(i) The number of combinations of n different things, taken k at a time without repetitions is the number of sets that can be made up from the n given things, each set containing k different things and no two sets containing exactly the same k things.
(ii) The number of combinations of n different things, taken k at a time, with repetitions is the number of sets that can be made up of k things chosen from the given n things, each being used as often as desired.
For Example: There are three combinations of the three letters a, b, c, taken two letters at a time, without repetitions, are ab, ac, bc, and six such combinations with repetitions, are ab, ac, bc, aa, bb, cc.
Theorem 4: The number of different combinations of n different things, k at a time, without repetition is
(4a)
The number of those combinations with repetitions is
Ex 5: The number of samples of five light bulbs that can be selected from a lot of 500 bulbs is
Theorem 5:
Or
Example 6: A student has seven books on his desk. In how many different ways can he select a set of three?
Solution: Since the order is not important, this is a combination problem:
Example 7. In how many ways can a committee of four be selected from a group of ten people?
Solution:
Example 8. Four cards are drawn at random from a pack of 52 cards. Find the probability that
(i) They are a king, a queen, a jack and a ace.
(ii) Two are kings and two are queens.
(iii) Two are black and two are red.
(iv) There are two cards of hearts and two cards of diamonds.
(8b) In shuffling a pack of cards, four are accidentally dropped, find the chance that the missing cards should be one from each suit.
Solution: Four cards can be drawn from a well-shuffled pack of 52 cards in 52C4 ways, which gives the exhaustive number of cases.
(i) 1 king can be drawn out of the 4 king in 4C1 ways. Similarly, 1 queen, 1 jack and an ace can each be drawn in 4C1 = 4 ways. Since any one of the ways of drawing a king can be associated with any one of the ways of drawing a queen, a jack and an ace, the favourable number of cases are 4C1x4C1x4C1x4C1.
Hence the required probability = = 256/52C4
(ii) Required probability =
(iii) Since there are 26 black cards (of spades and clubs) and 26 red cards (of diamonds and hearts) in a pack of cards, the required probability = .
(iv) Required probability =
(8b) Required probability =
Theorems on Probability
Theorem 1 If A is any event defined on finite sample space U, then
Where A’ is the complementary event of A.
Theorem 2
Theorem 3(a): Addition theorem of probability or theorem of Total Probability.
If A and B are two events which are not disjoint defined on a finite sample space U, then
Remark1. If A and B are mutually exclusive events (i.e. disjoint events) then
2. If A and B are mutually exclusive and exhaustive events, then and .
Theorem 3(b): If A, B, C are three events defined on finite sample space U, then
Example1. Two people are selected at random from a group of seven men and five women. Find the probability that both men or both are women.
Ans: 31/66
Example2. A card is drawn at random from a pack of 52 cards. What is the probability that the card is a spade or a king?
Ans: 4/13
Example 3. Two unbiased dice are tossed simultaneously. Find the probability that sum of numbers on the upper face of dice is 9 or 12.
Ans: 5/36
