Probability PDF

# PROBABILITY ## Introduction: Numerical study of chances of occurrence of events is dealt in probability theory. The theory of probability is applied in many diverse fields and the flexibility of the theory provides approximate tools for a great variety of needs. There are two approaches to pro...

# PROBABILITY ## Introduction: Numerical study of chances of occurrence of events is dealt in probability theory. The theory of probability is applied in many diverse fields and the flexibility of the theory provides approximate tools for a great variety of needs. There are two approaches to probability: 1. Classical approaches 2. Axiomatic approaches ## In both the approaches we use the term 'experiment', which when means an operation which can produce some well-defined outcome(s). There are two types of experiments: 1. **Deterministic experiment:** It an experiment, when repeated under identical conditions produce the Same result or outcome are known as deterministic experiments. When experiments in Science or engineering are repeated under identical conditions, we get almost the same result everytime. 2. **Random experiment:** It an experiment, when repeated under identical conditions, don't produce the same outcome every time but the outcome in a trial is one of the several possible outcomes then Such an experiment is known as a probabilistic experiment or a random experiment. ## In a random experiment, all the outcomes are known in advance but the exact outcome is unpredictable. For example, in tossing of a coin, it is known that either a head or a tail will occur but one isn't sure if a head or a tail will be obtained. So it is a random experiment. ## Definitions of various Terms: 1. **Sample Space:** The Set of all possible outcomes of a trial (random experiment) is called its Sample Space. It is generally denoted by S and each outcome of the trial is said to be a sample point. - Ex-(i) If a dice is thrown once, then its Sample Space is S={1,2,3,4,5,6} - (ii) If two coins are tossed together then its Sample Space is S = {HT, TH, TT, HH} 2. **Event:** An event is a subset of a sample Space. - i) **Simple event:** An event containing only a Single Sample Point is called an elementary or simple event. - Ex-In a Single toss of coins, the event of getting a head is a simple event - Here S = {H,T} and E = {H} - ii) **Compound events:** Events obtained by combining together two or more elementary events are known as the Compound events or decomposable events. - For example, In a single throw of a pair of dice the event of getting a doublet, is a compound event because this event occurs if any one of the elementary events (1,1), (2,2), (3,3), (4,4) (5,5), (6,6) Occurs. - iii) **Equally likely events:** Events are equally likely if there is no reason for an event to occur in preference to any other event. - Ex - If an unbiased die is rolled, then each outcome is equally likely to happen i.e. all elementary events are equally likely. - iv) **Mutually exclusive or disjoint events**: Events are Said to be mutually exclusive or disjoint or incompatible if the occurrence of any one of them prevents the occurrence of all the others - Ex- E = getting an even number - F = getting an odd number - These two events are mutually exclusive, because if E occurs we Say that the number obtained is even and so it cannot be odd i.e. F doesn't occur. - A₁ and A₂ are mutually exclusive events if A₁ ∩ A₂ = ø. - v) **Mutually non-exclusive events:** The events which are not mutually exclusive are known as Compatible events or mutually non-exclusive events. - vi) **Independent events:** Events are said to be Independent if the happening (or non-happening) of one event isn't affected by the happening (or non-happening) of others. - Ex - If two dice are thrown together, then getting an even no on first is independent to getting an Odd no on the Second. - vii) **Dependent events:** Two or more events are, said to be dependent if the happening of one event affects (partially or totally) other event. ## Note - Independent events are always taken from different experiments, while mutually exclusive events are taken from a single experiment. - Independent events can happen together while mutually exclusive events cannot happen together. - Independent events are connected by the word "and" but mutually exclusive events are connected by the word "or". ## Classical definition of Probability: It a random experiment results in n mutually exclusive, equally likely and exhaustive outcomes, out of which m are favorable to the occurrence of an event A, then the probability of occurrence of A is given by ``` P(A) = m/n = Number of outcomes favorable to A / Number of total outcomes. ``` It is obvious that 0 ≤ m ≤ n. If an event A is certain to happen, then m = n, thus P(A) = 1. If A is impossible to happen, then m = 0 and so P(A) = 0 Hence we conclude that 0 ≤ P(A) ≤ 1. Further, if 'A' denotes negative of A i.e. event that A doesn't happen, then for above cases we shall have ``` P(A) = n- m / n = 1- m/n = 1- P(A) .• P(A) + P(A) = 1. ``` ## Notations: For two events A and B, - i) A or A or A stands for the non-occurrence or negation of A. - ii) AUB stands for the occurrence of at least one of A and B. - iii) ANB stands for the simultaneous occurrence of A and B. - iv) 'A'n'B' Stands for the non-occurrence of both A and B. - v) A⊂B stands for "the occurrence of A implies occurrence of B." ## Some important remarks about coins, Dice, playing cards and Eyelups: 1. **Coins:** A coin has a head side and a tail side. It an experiment consists of more than a coin, then coins are considered to be distinct if not otherwise stated. - Number of exhaustive cases of tossing n coins Simultaneously (or of tossing a coin n times) = 2ⁿ 2. **Dice:** A die (Cubical) has 6 faces marked 1,2,3, 4, 5,6. We may have tetrahedral (having four faces 1, 2, 3, 4} or Pentagonal (having five faces 1, 2, 3, 4, 5) die. As in the case of coins, if we have more than one die, then all dice are considered to be distinct if not otherwise stated. - Number of exhaustive cases of throwing n dice Simultaneously (or throwing one dice n times) = 6ⁿ 3. **Playing Card:** A pack of playing cards usually has 52 Cards. There are 4 Suits (Spade, Heart, Diamond and club) each having 13 cards. There are two colours red (Heart and Diamond) and black (Spade and club) each having 26 cards. - In thirteen cards of each suit, there are 3 face cards or court cards namely King, queen and jack so there are in all 12 face cards (4 Kings, 4 queen and 4 jacks). Also there are 16 honor cards, 4 of each suit namely ace, King, queen and Jack ## 4) Probability regarding n letters and their envelopes: If n letters corresponding to n envelopes are placed in the envelopes at random, then - i) Probability that all letters are in right envelopes = 1/n! - ii) Probability that all letters aren't in right envelopes = 1 - 1/n! - iii) Probability that no letters is in right envelopes = 1/n! - 1/n - iv) Probabilty that exactly r letters are in right envelopes = 1/r! [1/n! + 1/n(n-1)! + (n-r)! ] / (n-r)! ## Problems based on combination & Permutation: To solve such kind of problems, we use nCr = n! / r!(n-r)! ## Odds In Favour & odds against an Event: As a result of an experiment if "a" of the outcomes are favourable to an event E and "b" of the outcomes are against it, then we say that odds are a to b in favour of E of odds are b to a against E. Thus odds in favour of an event E = No. of favourable cases / No. of unfavourable cases = a / (a+b) = P(E) / 1-P(E) Similarly, odds against an event E = No. of unfavourable case b / No. of favourable case a = P(E) / 1-P(E) ## Long questions: 1. Given P(A) = 1/3 and P(B) = 1/2, Find P(A or B), if A and B are mutually exclusive events. 2. If E and F are events such that P(E) = 1/4, P(F) = 1/3 and P(E and F) = 1/8, find (i) P (E or F) (ii) P (not E and not F) 3. Events E and F such that P (not E or not F) = 0.25. State whether E and F are mutually exclusive. 4. In class XI of a School 40% of the students study Mathematics and 30% Study Biology. 10% of the class study both Mathematics and Biology. If a student is selected at random from the class, find the probability that he will be studying Mathematics or Biology. 5. In an entrance test that is graded on the basis of two examinations, the probability of a randomly chosen student passing the first examination is 0.8 and the probability of passing the second examination is 0.7. The probability of pawing atleast one of them is 0.95. What is the probability of paving both? 6. The number lock of suitcase has 4 wheels, each labelled with ten digits i.e. from 0 to 9. The lock opens with a sequence of four digits with no repeats. What is the Probability of a person getting the right sequence to open the Quitcase? ## Objective questions: 1. Two Cards are drawn at random from a deck of 52 Cards. The probability of these two being aces is - (a) 1/26 - (b) 7/221 - (c) 1/2 - (d) 1/18 2. A card is drawn from a well-shuffled deck of cards. The probability of getting a queen of club or king of heart is - (a) 1/52 - (b) 1/26 - (c) 1/13 - (d) 1/56 3. In shuffling a pack of cards 3 are accidently dropped, then the the chance that missing card should be of different Suits is - (a) 169/425 - (b) 26/425 - (c) 104/425 - (d) 169/425 4. If there are 4 addressed envelopes and 4 letters. Then, the chance that all the letters are not mailed through Proper envelope is - (a) 1/24 - (b) 1 - (c) 23/24 - (d) 9/24 5. A and B are two events Such that P(A)=0.3 and P(AUB)=0.8. If A and B are Independent, then P(B) is - (a) 2/3 - (b) 3/8 - (c) 2/7 - (d) 5/7 6. A speaks truth in 60% cases & B speaks truth in 70% cases. The probability that they will say the same thing while describing Single events, is - (a) 0.56 - (b) 0.54. - (c) 0.38 - (d) 0.94 7. The probability that in the toss of two dice, we obtain an even sum or a sum less than 5 is - (a) 1/2 - (b) 1/6 - (c) 1/3 - (d) 5/9 8. The probabilities of Solving a problem by three students, A,B and C are 1/2, 1/3 and 1/4 respectively. The probability that the problem will be solved is - (a) 1/4 - (b) 1/2 - (c) 3/4 - (d) 1/3 9. The Probability that a man can hit a target is 3/4. He tries 5 times. The probability that he will hit the target atleast three times is - (a) 371/364 - (b) 471/464 - (c) 502/512 - (d) 459/512 10. A draws two cards with replacement from a deck of 52 Cards and B throws a pair of dice. The chance that A gets both cards of Same Suit and B gets total of 6 is - (a) 1/144 - (b) 1/4 - (c) 5/144 - (d) 7/144 11. If A and B are such events that P(A) > 0 and P(B) ≠ 1, then P(A/B) is equal to - (a) 1 - P(AUB) / P(B) - (b) 1 - P(A/B) / P(B) - (c) 1- P(A ∩ B) / P(B) - (d) P(A) / P(B) 12. In solving any problem, odds against A are 4 to 3 and odds in favour of B in solving the same is 7 to 5. Then, Probability that problem will be solved is - (a) 5/21 - (b) 16/21 - (c) 15/84 - (d) 69/84 13. Two events, A and B have probability 0.25 and 0.5. The probability that both occur simultaneously is 0.14. Then, probability that neither A nor B occur is - (a) 0.75 - (b) 0.61 - (c) 0.39 - (d) None of these. 14. A contest consists of predicting the results (win, draw or defeat) of 8 matches played by the Indian cricket team. A person sent his entry by predicting at random. The probability - (a) 1/2^8 - (b) 1/3^8 - (c) 1/1120 - (d) 3/10^2 15. Of cigarette Smoking population 70% are men and 30% are women, 10% of these men and 20% of these women Smoke wills. The Probability that a person Seen Smoking a wills to be men is - (a) 1/5 - (b) 7/13 - (c) 5/13 - (d) 7/10 16. Two friends P and Q have equal number of daughters. The two friends have three cinema tickets which are to be distributed among their daughters. If the probability that all the tickets go to daughters of P be 1/20, then the no of daughters each has is - (a) 5 - (b) 3 - (c) 4 - (d) 6 17. A Cricket team has 15 members, of whom only 5 can bowl. If the names of the 15 members are put into a bat and 11 drawn at random, then the chance of obtaining an eleven containing atleast 3 bowlers is - (a) 7/13 - (b) 11/15 - (c) 12/13 - (d) 15/14 18. The probability of India winning a test match against England is 2/3. Assuming independence of the result of various matches, the chance that in a 5 match series, India's second win occur at 3rd text is - (a) 2/3 - (b) 2/7 - (c) 2/9 - (d) 1/2 19. There is a point inside a circle. what is the probability that this point is closer to the circumference than to the centre ? - (a) 3/4 - (b) 1/2 - (c) 1/4 - (d) 1/3 20. A husband and wife appear in an interview for two vacancies in the same post. The probability of husband's selection is 1/2 and that of wife Selection is 1/3. What is the probability that only one of them will be selected? - (a) 1/4 - (b) 2/3 - (c) 3/5 - (d) 4/5