“PROBABILITY”

Some important terms Event: An event is one or more of the possible outcomes of an activity. When we toss a coin there are two possibilities, Head or Tail. Both these are the events. The flipping of a coin is the activity. Experiment: An act which can be repeated under some given conditions. Random Experiment: If in an experiment, all the possible outcomes are known in advance and none of the outcomes can be predicted with certainty. Eg. Tossing of a coin, Throwing of a die.

Sample Space: The sample space is a set of all possible outcomes of an experiment. S = [Head, Tail] When we toss a coin once. Mutually Exclusive Events: If one and only one event takes place at a time they are mutually exclusive events. For e.g. Head and Tail are mutually exclusive events Exhaustive Events: These events present all the possible outcomes of an experiment; head and tail are collectively exhaustive

Dependent Events: If the occurrence of one event affects the probability of occurrence of the other event, then the two events are said to be dependent events. Independent Events: When the occurrence of one event does not affect the probability of the occurrence of the other event, then the two events are said to be independent events.

Probability or Chance The probability is an expression of likelihood or chance of Occurrence of an event. Probability is a number which ranges from 0 to 1 0  an event cannot occur. 1  an event is certain to occur.

Classical or Priori Probability The probability of occurrence of an event A, denoted by p(A) is given as p (A) = No. of favorable cases / Total no. of equally likely cases Example: From a bag containing 10 black and 20 white balls, a ball is drawn at random. What is the probability that it is black.

Theorems of Probability: Addition Theorem Multiplication Theorem Addition Theorem: If two events A and B are mutually exclusive events, the probability of occurrence of either A or B is the sum of Individual probabilities of A and B. i.e. p ( A or B) = p(A) + p(B) Or p ( A U B) = p(A) + p(B)

Example: One card is drawn at random from the pack of 52 cards. What is the probability that it is either a king or a queen. Some facts about Playing cards Total number of cards : 52 Total number of red cards : 26 Total number of black cards : 26 Red cards : 13 hearts + 13 diamonds Black cards : 13 spades + 13 clubs 4 (2R+2B) Ace, 4 (2R+2B) King, 4 (2R+2B) Queen, 4 (2R+2B) Knave or Jack 4 (2R+2B) Numbers from 2 to 10

When events are not mutually exclusive Then the modified form of Addition Theorem is p ( A or B) = p (A) + p (B) – p (A and B) i.e. p (A U B) = p (A) + p (B) – p (A  B) Example: A bag contain 30 balls, numbered from 1 to 30, a ball is drawn at random. Find the probability that the number of the ball drawn will be a multiple of (a) 5 or 7 (b) 3 or 7 A card is drawn at random from a pack of 52 cards what is the probability that it is either a Knave or a spade.

Multiplication Theorem: If two events A and B are independent, the probability that they Both will occur is equal to the product of their individual Probabilities. i.e. p (A and B) = p (A) X p (B) p(A and B and C) = p(A) X p(B) X p(C) Example: A man wants to get married. The girl whom he wants to marry should have the following qualities: 1.White Complexion, the probability of getting such girl is 1/20 2. Handsome Dowry, the probability is 1/50 3. Westernized manner, the probability is 1/100 What is the probability that this man will get married?

Conditional Probability: ( for Dependent events) Example: A bag contains 5 White and 3 Black balls. Two balls are drawn At random one after another without replacement. Find the probability that both the balls are black. Example: Find the probability of drawing a Queen, a King, and a Knave In that order from a pack of cards in three consecutive draws, When the cards were 1.Not replaced 2.Replaced.

Combination Number of combinations of n objects taken r at a time Is denoted by n C r = n! / r! (n – r)! Here n! = n(n -1) (n –2) (n –3)………1 Example: A bag contains 6 White, 4 Red and 10 Black balls. Two balls are drawn at random. Find the probability that they both will be black.

Example: A bag contains 8 White, 4 Red balls. Five balls are drawn at random. What is the probability that 2 are Red And 3 are White. Example: From a pack of 52 cards, two cards are drawn at random. Find the probability that one is King and one is Queen. Example: One bag contains 4 White and 2 Blue balls. Another bag contains, 3 White and 5 Blue balls. If one ball is drawn from each, find probability that, (a) Both are White (b) Both are Blue (c) One is White and other is Blue

Baye’s Theorem: Given by British mathematician Mr. R.T. Bayes It is an extension of conditional probability. Let, A1 and A2 be the set of events which are mutually exclusive and exhaustive. And B is a simple event which intersects each the events A1 and A2 as BA1A2

Then the Probability of event A1 when B is given is denoted as p(A1/B) = p(A1 and B) / p(B) And similarly the probability of event A2, given B is p(A2/B) = p(A2 and B) / p(B) Where p(B) = p( A1 and B) + p( A2 and B) Also p( A1 and B) = p(A1) X p(B/A1) p( A2 and B) = p(A2) X p(B/A2)

1 Events 2 Prior Probability P(Ai) 3 Conditional Probability event B given event A P(B/Ai) 4 Joint Probability 2x3 5 Posterior Probability 4 / P(B) A1 A2. Total1.00P(B)1.00 Computation of Posterior Probabilities

Example: Assume that factory has two machines. Past records show that machine 1 produce 30% of the items of output and machine 2 produces 70% of the items. Further 5% of the items produced by the machine items. Further 5% of the items produced by the machine 1 were defective and only 1% of the items produced by machine 2 were defective. If a defective item is drawn at random, what is the probability that the defective item was produced by machine 1.

Example: A manufacturing firm produces units of products in four plants. Define; Ai - a unit is produced in plant i( i= 1, 2, 3, 4) and event B - a unit is defective. From the past records of the proportion of defectives produced As, p(B/A1) = 0.05, p(B/A2) = 0.10, p(B/A3) = 0.15, p(B/ A4) = 0.02 The first plant produces 30% of the units of the product, The second plant produces 25% of the product, The third plant produces 40% of the product and The fourth plant produces 5% of the product. A unit of the product made at one of these plants is tested and found to be defective. What is the probability that the unit was produced in plant 3.

Questions for practice: Q1. A bag contains 5 white and 8 red balls. Two drawings of 3 balls are made in such a manner (a)The balls are replaced (b) The balls are not replaced before the second trial. Find the probability that the first drawing will give 3 white and The second will give 3 red balls in each case. Q.2 A university has to select an examiner from a list of 50 persons. 20 of them are women and 30 men. 10 of them know Hindi and rest 40 do not, 15 of them are in teaching Profession. What is the probability of the university to select A hindi knowing Woman Teacher.

Contd.. Q.3 A fair dice is tossed twice. What is the probability of getting 4, 5, or 6 on the first toss and 1, 2, 3, or 4 on the second toss. Q.4 What is the probability that a leap year selected at random will contain 53 Sundays. Q.5 In a bolt factory machine A, B, C manufacture 25%, 35%, and 40% of the product. Out of the total of their output 5,4, and 2% are defective. A bolt is drawn at random and is found to be Defective. Find the probability that it was manufactured by machine A, B and C.