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PROBABILITY THEORY. T Introduction to Probability T heory. History and Relevance of probability theory Probability theory began with the study of game of chance that were related to gambling, like throwing a die, drawing a card from a deck of card, tossing a coin , etc.
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TIntroduction to Probability Theory • History and Relevance of probability theory Probability theory began with the study of game of chance that were related to gambling, like throwing a die, drawing a card from a deck of card, tossing a coin , etc.
Some basic Definition of Probability • Random Experiment any process that generates a set of data is referred to as an experiment . If an experiment is repeted under essentially identical conditions and the possible outcomes are not unique but can be any of the prior described outcomes then such an experiment is called a random experiment.
EXAMPLE: Tossing of a coin, throwing a die leads to six possible outcomes-getting any of the numbers 1,2,3,4,5,6
Trial and Event When a real or conceptual experiment is repeated under Essentially the same conditions, we obtain a set of possible outcomes. Then each experiment is called a trial and the possible outcomes is known as event or cases.
EXAMPLE: Throwing a die is a trial and getting 1,2,3,4,5,6 is known as an event.
Equally Likely Cases Events are said to be equally likely or equally possible if the likelihood of the occurrence of every event is same.
EXAMPLE: In tossing an unbiased coin , the chances of the turning up of the head or tail is equal
Mutually exclusive event • If the occurrence of one case precludes the occurrence of the other, then they are called as mutually exclusive or incompatible cases.
EXAMPLE: In tossing of a coin we observe the fact that the turning up of head precludes the turning up of tail and vice-versa.
Sample Space and Algebra of Events • Sample space – The set of all possible outcomes of a random experiment is called the sample space and it is represented by the symbol ‘s’
Sample point • Each outcome in a sample space is called an element or member of the sample space or simply a sample point.
Example: If we tossing a coin , the sample space S can be written as S= {H,T}
Where H and T denote the occurrence of ‘Head’ and ‘Tail’ some time it is better to list the elements of a sample space in an organized manner by means of a ‘Tree Diagram’
Illustration1- : list all the possible sample points by using a tree diagram of an experiment of tossing a coin and then tossing it again if a head occurs . If a tail occurs on the first toss , then a die is tossed once. Solution: To list the element of the sample space providing the most information, we construct the tree diagram as shown in the following fig.
TREE DIAGRAM: SECOND OUTCOME SAMPLE POINT FIRST OUTCOME T HH HT T1 T2 T3 T4 T5 T6 H H START 1 2 T 3 4 5 6
NOTE- We see from the diagram that the distinct sample points are given by the various paths along the branches of the tree. Starting with the top left branch, moving to the right along the first path, we get the sample point HH which indicates the possibility that head occurs on two successive tosses of the coin. Thus proceeding along all paths, we see that the sample space is: S= {HH,HT,T1,T2,T3,T4,T5,T6}
VENN DIAGRAM: The relationship between events and the corresponding sample space can be illustrated graphically by venn diagram.
IMPORTANT: • IN A VENN DIAGRAM, THE SAMPLE SPACE IS REPRESENTED BY A RECTANGLE • THE EVENTS ARE REPRESENTED BY CIRCLES • CIRCLE ARE DRAWN INSIDE THE RECTANGLE.
A B 2 7 6 1 3 4 5 C
A∩B = Region 1 and 2 • B∩C = Region 1 and 3 • AUC = Region 1,2,3,4,5 and 7
A∩BUC = Region 1 • B´∩A = Region 4 and 7 • (AUB)∩C´
S A C B
From the above fig. it is clear that: • Event B is a subset of event A; • Event B∩C has no elements; • Event A∩C has atleast one element;
ILLUSTRATION: Let A and B be two events. Exhibit by a Venn diagram the event that A but not B occurs. Exactly one of the two events occurs.
SOLUTION: S (I) B A
(II). S B A
Permutations: An arrangement of a set of ‘n’ objects in a given order is called a permutation of the objects (taken all at a time). Illustration: Consider the three letters x ,y , z .the possible permutations are xyz , xzy,yxz,yzx, zxy, and zyx. Here we see that there are 6 distinct arrangements.
In above: If n₁=3, n₂=2, n₃=1 Then n₁ n₂ n₃ = (3) (2) (1) =6 permutations
A Permutation of ‘n’ objects taken ‘r’ at a time is given by:
A Permutation of ‘n’ objects taken ‘r’ at a time is given by:
Combinations: Combinations of ‘n’ different objects taken ‘r’ at a time is a selection of ‘r’ out of ‘n’ objects where the order is not considered. This is denoted by
ILLUSTRATION: From 4 men and 6 women, find the number of committees of 3 that can be formed with 2 men and 1 woman.
SOLUTION: The number of ways of selecting 2 men out of 4 is, ⁴c₂ =4!/2!x2! =4x3x2x1/2x1x2x1 =4x3/2 =6
Similarly 1 woman out of 6 can be selected in ⁶c₁ = 6! / 1! x 5! =6 ways Now using the multiplication rule, the number of committees that can be formed is, 6 x 6 = 36 Result is - 36
Define the following terms as used in probability: • random experiment • Sample space • Event • Union of two events • Intersection of two events
Differentiate between independent and mutually exclusive events. Can two event be mutually exclusive and independent simultaneously? • What is the probability of throwing an even number with a die ?
What is chances of throwing a 4 with an ordinary die? • A coin is weighted so that head is twice as likely to appear as tails. Find P(T) and P(H) ? • An urn contains 10 red and 3 white balls. Find the probability that if two balls are drawn out of it, then they are white?
If the letters of the word ‘SPECULATION’ be arranged at random, what is the chances that there will be exactly 4 letters between ‘S’ and ‘P’ ? • A bag contain 7 red, 12 white and 4 blue balls. What is the probability that three balls drawn at random are one of each colour?
