Probability
Definition:
Probability is the study of random or nondeterministic experiments.
• (The term ‘experiment’ refers to describe an act which can be repeated under some given conditions)
A random or statistical experiment is one in which
(i) all possible outcomes of the experiment are known in advance.
(Outcome: possible result of an experiment)
(ii) a performance of an experiment results in an outcome which is not known in advance.
(iii) the experiment can be repeated under identical conditions.
Ex-1: If we toss a coin, then it results either in a head or a tail; of course we can’t be sure of the outcome in advance.
Ex-2: If we roll a dice, then it results either 1 or 2 or 3 or 4 or 5 or 6; of course we can’t be sure of the outcome in advance.
• If we toss a coin, we are sure to get either a head (h) or a tail (t); that is either ‘h’
occurs or ‘t’ occurs. In the language of probability, the set
is called the sample space of the experiment (of tossing a coin).
• If we throw a dice, then we are sure to get one of these results: 1,2,3,4,5,6. The set
is called the sample space of the experiment (of throwing a dice).
Sample space: The sample space of an experiment is the set of all possible outcomes of the experiment.
• A particular outcome, i.e. an element in S, is called a sample point or sample.
• When we repeat a random experiment several times, we call each one of them is a trail.
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• Ex-1: Toss a coin twice. The possible outcomes are hh, ht, th, tt.
Therefore the sample space is
• Ex-2: Toss a coin thrice. In this case
• Ex-3: Throw a dice twice. There are 36 possible ways.
Sample sets are finite in above examples.
All sample sets are not finite.
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• Ex-4: One is allowed to go on tossing a coin until a head is obtained. In this case the sample space is
which is not finite.
• Ex-5: Pick up a integer at random from the set of positive integers. In this case the sample space is
which is an infinite set.
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Definition: Event
A subset of a sample space (of an experiment) is called an event. An event is said to occur if an element of the event occurs.
• The event consisting of a single space is called an elementary event.
1. itself is an event.
2. The empty set is an event.
We can combine events to form new events using the various set operations:
3. If and are events (i.e. ) then is an event.
4. Similarly is an event that occurs iff occurs and occurs.
5. If is an event is an event that occurs iff does not occur.
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• Ex-6 (From Ex-3):
Let which is a subset of Clearly, can be described as the event that sum of points obtained in two throws of dice is 7.
• Ex-7 (From Ex-3):
Let which is a subset of Clearly, can then be described as the event that the result of the first throw of the dice exceeds the result of the second throw by 1.
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Probability of an event
• If you toss a coin once, sample space The chance/probability of getting a head is and the chance/probability of getting a head is .
In shorts, and
• If you toss a coin twice, sample space Let i.e. the event of getting one head and one tail. Then will occur twice out of four times, thus .
• Definition: If the sample space is finite, then the probability of an event denoted by is defined as
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Different types of events:
1. Mutually Exclusive Events: Two events are said to be mutually exclusive when both cannot happen simultaneously in a single trial.
If and are mutually exclusive events, i.e. when .
• For example, if a single coin is tossed either head can be up or tail can be up, both cannot be up at the same time.
• Similarly, a person may be either alive or dead at a point of time, he cannot be both alive as well as dead at the same time.
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2. Independent and Dependent Events:
Two or more events are said to be independent when the outcome of one does not affect, and is not affected by the other.
For example, if a coin is tossed twice, the result of the second throw would in no way be affected by the result of the first throw.
Similarly, the results obtained by throwing a dice are independent of the results obtained by drawing an ace from a pack of cards.
Dependent events are those in which the occurrence or non-occurrence of one event in any one trial affects the probability of other events in other trials.
For example, the probability of drawing a queen from a pack of 52 cards is . But if the card drawn (queen) is not replaced in the pack, the probability of drawing again a queen is .
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3. Simple and Compound Events
In case of simple events we consider the probability of happening or not happening of single events.
For example, we might be interested in finding out the probability of drawing a red ball from a bag containing 10 white and 6 red balls.
On the other hand, in case of compound events we consider the joint occurrence of two or more events.
For example, if a bag contains 10 white and 6 red balls and if two successive draws of 3 balls are made, we shall be finding out the probability of getting 3 white balls in the first draw and 3 red balls in the second draw- we are thus dealing with a compound event.
4. Exhaustive Events
Events are said to be exhaustive when their totality includes all the possible outcomes of a random experiment.
For example, while rolling a dice, the possible outcomes are 1, 2, 3, 4, 5 and 6 and hence the exhaustive number of cases is 6. If two dice are thrown once, the possible outcomes are:
The sample space of the experiment i.e. 36 ordered pairs ().
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5. Complementary Events:
• Let there be two events and . is called the complementary event of (and vice versa) if and are mutually exclusive and exhaustive.
• For example, when a dice is thrown, occurrence of an even number and odd number are complementary events.
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6. Equally Likely Events
Events are said to be equally likely when one does not occur more often than others.
For example, if an unbiased coin or dice is thrown, each face may be expected to be observed approximately the same number of times in the long run.
(Unbiased coin: has equal probability of heads or tails Biased coin: has a higher probability of heads or tails)
Rules/Axioms of Probability
• Let be a sample space, be the class of events, and be a real-valued function defined on . Then is called a probability function and is called the probability of event if the following axioms holds:
1. For every event ,
2. If and are mutually exclusive events, i.e. when , then
In general, if are mutually exclusive, then
4. In general, if is a sequence of mutually exclusive events, then
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Theorems:
1. If is the empty set, then
2. If is the complement of an event , then 3. If are any two events and , then
4. For any two events ,
5. If are two events, then
6. If two events are independent, the probability that they both will occur is equal to the product of their individual probability.
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Corollary:
For any events , then
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Example -1: Find the probability of getting an even number in throwing a dice.
Example-2: A dice is throw twice. Find the probability that the sum of points obtained is 8.
Example-3: Two cards are drawn from a standard pack of 52 cards. Find the probability that both cards are aces.
Example-4: A bag contains 5 white and 3 black balls. If a ball is drawn at random, find the probability that it is white.
Example-5: A bag contains 6 white and 7 black balls. If two balls are drawn at random, find the probability that both balls are white.
Example-6:Two balls are drawn from a bag containing 4 white and 6 black balls. Find the probability that at least one of the balls is white.
Example-7:Let a card be selected at random from an ordinary deck of 52 cards. Let
and .
Compute , and .
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Example-8: Let 2 items be chosen at random from a lot containing 12 items of which 4 are defective. Let
and
Find and .
What is the probability that at least one item is defective?
Example-9:Two cards are drawn at random from an ordinary deck of 52 cards. Find the probability that (i) both are spades (ii) one is a spade and one is a heart.
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Example-10: Two cards are selected at random from 10 cards numbered 1 to 10. Find the probability that the sum is odd if (i) the two cards are drawn together, (ii) the two cards are drawn one after the other without replacement, (iii) the two cards are drawn one after the other with replacement.
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Example 11: The managing committee of Shantivan Welfare Association formed a sub-committee of 5 persons to look into electricity problem.
Profiles of 5 persons are:
1. Male age 40 2. Male age 43 3. Female age 38 4. Female age 27 5. Male age 65
If a chairperson has to be selected from this, what is the probability that he would be either female or over 30 years?
Example 12: A person is known to hit the target in 3 out of 4 shots, whereas another person is known to hit the target in 2 out of 3 shots.
Find the probability of the target being hit at all when they both try.
Example 13: Calculate the probability of picking a card that was a heart or a spade.
Example 14: What is the probability of picking a card that was red or black?
Example 15: A bag contains 30 balls numbered from 1 to 30. One ball is drawn at random. Find the probability that the number of the ball drawn will be a multiple of (a) 5 or 7, and (b) 3 or 7.
Example 16: A problem in statistics is given to five students A, B, C, D and E. Their chances of solving it are and . What is the probability that the problem will be solved?
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Example 17: Two students X and Y work independently on a problem. The probability that X will solve it is and the probability that Y will solve it is What is the probability that the problem will be solved?
Example 18: A bag contains 4 white, 3 black and 5 red balls. What is the probability of getting a white or a red ball at random in a single draw?
Example 19: In two tosses of a coin, what are the chances of head in both?
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• Example 20: What is the probability that a randomly chosen card from a deck of cards will be either a king or a heart?
Example-21: Of 1000 assembled components, 10 have a working defect and 20 have a structural defect. There is a good reason to assume that no component has both defects. What is the probability that randomly chosen component will have either type of defect?
Example 22: From a sales force of 150 persons, 1 will be selected to attend a special sales meeting. If 52 of them are unmarried, 72 are college graduates, and 3/4 of the 52 that are unmarried are college graduates, find the probability that the sales person selected at random will be neither single nor a college graduate.
Example 23: From a computer tally based on employer records, the personnel manager of a large manufacturing firm finds that 15 per cent of the firm’s employees are supervisors and 25 per cent of the firm’s employees are college graduates. He also discovers that 5 per cent are both supervisors and college graduates. Suppose an employee is selected at random from the firm’s personnel records, what is the probability of (a) selecting a person who is both a college graduate and a supervisor?
(b) selecting a person who is neither a supervisor nor a college graduate?
Example 24: The probability that a contractor will get a plumbing contract is 2/3 and the probability that he will not get an electrical contract is 5/9. If the probability of getting at least one contract is 4/5, what is the probability that he will get both?
Example 25: An MBA applies for a job in two firms X and Y. The probability of his being selected in firm X is 0.7 and being rejected at Y is 0.5. The probability of at least one of his applications being rejected is 0.6. What is the probability that he will be selected by one of the firms?
Example 26: A husband and wife appear in an interview for two vacancies in the same post. The probability of husband’s selection is 1/7 and that of wife’s selection is 1/5. What is the probability that (a) both of them will be selected, (b) only one of them will be selected, and (c) none of them will be selected.
Conditional Probability: (Dependent Events)
• Conditional probability is the probability of one event occurring with some relationship to one or more other events.
• Let be an arbitrary event in a sample space with . The probability that an event occurs, once has occurred or, in other words, the conditional probability of given , written , is defined as follows:
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Example: Suppose a bag contains 6 balls, 3 red and 3 white. Two balls are chosen (without replacement) at random, one after the other. Consider the two events
is event “first ball chosen is red”
is event “second ball chosen is white”
We easily find However, determining the probability of is not quite straight forward. If the first ball chosen is red then the bag subsequently contains 2 red and 3 white balls. In this case, However, if the first ball chosen is white then the bag subsequently contains 3 red and 2 white balls. In this case,
What this example shows is that the probability that occurs is clearly dependent upon whether or not the event has occurred. The probability of occurring is conditional on the occurrence of
The conditional probability of an event occurring given that event has occurred is written as
In this particular example, and
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Example:
Suppose if you roll a dice,
Suppose we are told that the dice came up with an even number, no other information.
Reduced sample space,
Given an even number is rolled, what is the probability that it is a 2?
So the probability of getting a 2, given we rolled an even number is 1 out of 3.
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Theorem:
Let be a finite equiprobable space with events and . Then
or,
• A sample space S is called an equiprobable space if and only if all the simple events are equally likely to occur.
• e.g., A toss of a coin. It is equally likely for a head to show up as it is for a tail. Throwing a die. Each number on the die has the same amount of chance of coming up.
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Multiplication Theorem for Conditional Probability
If and are any two events,
For any events ,
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Example: Let a pair of fair dice be tossed. If the sum is 6, find the probability that one of the dice is a 2. In other words, if
and Find
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Example: A company has two plants to manufacture scooters. Plant I manufactures 80 per cent of the scooters and Plant II manufactures 20 per cent. In plant I, only 85 out of 100 scooters are considered to be of standard quality. In plant II, only 65 out of 100 scooters are considered to be of standard quality. What is the probability that a scooter selected at random came from plant I, if it is known that it is of standard quality?
• Let A = The scooter purchased is of standard quality;
B = The scooter is of standard quality and came from plant I;
C = The scooter is of standard quality and came from plant II;
D = The scooter came from plant I.
The percentage of scooters manufactured in plant I that are of standard quality is 85 per cent of 80 per cent, that is, 0.85 × (80 ÷ 100) = 68 per cent or P(B) = 0.68.
The percentage of scooters manufactured in plant II that are of standard quality is 65 per cent of 20 per cent, that is, 0.65 × (20 ÷ 100) = 13 per cent or P(C) = 0.13.
The probability that a customer obtains a standard quality scooter from the company is 0.68 + 0.13 = 0.81. The probability that the scooters selected at random came from plant I, if it is known that it is of standard quality, is given by
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Example: The probability that a trainee will remain with a company is 0.6. The probability that an employee earns more than Rs. 10,000 per month is 0.5. The probability that an employee who is a trainee remained with the company or who earns more than Rs. 10,000 per month is 0.7. What is the probability that an employee earns more than Rs. 10,000 per month given that he is a trainee who stayed with the company?
Example: Two computers A and B are to be marketed. A salesman who is assigned the job of finding customers for them has 60 per cent and 40 per cent chances of succeeding for computers A and B, respectively. The two computers can be sold independently. Given that he was able to sell at least one computer, what is the probability that computer A has been sold?
• Example: Suppose that one of the three men, a politician, a bureaucrat and an educationist will be appointed as the Vice-Chancellor of a University. The probabilities of appointment are respectively 0.3, 0.25 and 0.45. The probability that research activities will be promoted by these people if they are appointed is 0.4, 0.7 and 0.8 respectively. What is the probability that research will be promoted by the new Vice- Chancellor?
Bayes’ Theorem
Suppose the events form a partition of a sample space i.e., the events are mutually exclusive and their union is . Now let be any other event. Then,
Where, are also mutually exclusive. Accordingly
Thus by multiplication theorem, ---(1)
On the other hand, for any , the conditional probability of given is defined by In this equation, we use (1) to replace and use to replace , thus obtaining.
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Suppose is a partition of and is any event. Then for any ,
• ; where are events and .
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how often A happens given that B happens, written P(A|B), how often B happens given that A happens, written P(B|A)
and how likely A is on its own, written P(A) and how likely B is on its own, written P(B)
• Example: Three machine A, B and C produce respectively 60%, 30% and 10% of the total number of items of a factory. The percentages of defective output of these machines are respectively 2%, 3% and 4%. An item is selected at random and is found defective. Find the probability that the item was produced by machine C.
• Example: Three machine A, B and C produce respectively 50%, 30% and 20% of the total number of items of a factory. The percentages of defective output of these machines are 3%, 4% and 5%. An item is selected at random and is found defective. Find the probability that the item was produced by machine A; that is, find
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• Example: In a certain college, 4% of the men and 1% of the women are taller than 6 feet. Furthermore, 60% of the students are women. Now if a student is selected at random and is taller than 6 feet. What is the probability that the student is a women?
• Example: In a certain college 25% of the students failed in mathematics, 15% of the students failed in chemistry, and 10% of the students failed in both mathematics and chemistry. A student is selected at random:
1. If he failed in chemistry, what is the probability that he failed in mathematics.
2. If he failed in mathematics, what is the probability that he failed in chemistry.
3. What is the probability that he failed in mathematics or chemistry?
• Example: A company has two plants to manufacture scooters. Plant I manufactures 70% of the scooters and plant II manufactures 30%. At plant I, 80% of scooters are rated standard quality and at plant II, 90% of scooters are rated standard quality. A scooter is picked up at random and is found to be of standard quality. What is the chance that it has come from plant I ? What is the chance that it has come from plant II?
• Example: The probabilities of , and becoming managers are and respectively. The probability that the bonus scheme will be introduced if and become managers are , and respectively.
a) What is the probability that the bonus scheme will be introduced?
b) If the Bonus scheme has been introduced, what is the probability that the manager appointed was
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Some Additional Problem
Example: Let and be events with , and Find
Example: Let and be events with , and Find, and .
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• Example: A lot contains 12 items of which 4 are defective. Three items are drawn at random from the lot one after the other. Find the probability that all three are non-defective.
• Example: A class has 12 boys and 4 girls. If 3 students are selected at random from the class, what is the probability that they are all boys?
• Example: A man is dealt 5 cards one after the other from an ordinary deck of 52 cards. What is the probability that they are all spades?
• Example: An urn contains 7 red marbles and 3 white marbles. Three marbles are drawn from the urn one after the other. Find the probability that the first two are red and the third is white.
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• Example: A university has to select an examiner from a list of 50 person, 20 of them women and 30 of men, 10 of them knowing Hindi and 40 not. 15 of them being teachers and the remaining 35 not. What is the probability of university selecting a Hindi-knowing women teacher?
• Example: The personal department of a company has records which show the following analysis of its 200 engineers:
Educational qualifications
Age Bachelor's Degree only Master Degree Total
Under 30 90 10 100
30 to 40 20 30 50
Over 40 40 10 50
Total 150 50 200 If one engineer is selected at random from the company, find:
i. the probability that he has only a bachelor’s degree,
ii. the probability that he has a master degree, given that he is over 40,
iii. the probability that he is under 30, given that he has only a bachelor’s degree.
