Probability A quantitative measure of uncertainty A quantitative measure of uncertainty A measure of degree of belief in a particular statement or problem A measure of degree of belief in a particular statement or problem Probability is a measure of how likely it is for an event to happen. Probability is a measure of how likely it is for an event to happen. The probability and statistics are interrelated The probability and statistics are interrelated Foundation of Probability were laid by two French Mathematician, Blaise Pascal, Pierre De Fermat Foundation of Probability were laid by two French Mathematician, Blaise Pascal, Pierre De Fermat Probability theory has a wider field of application and is used to make intelligent decision in Management, Economics, Operation Research, Sociology, Psychology etc Probability theory has a wider field of application and is used to make intelligent decision in Management, Economics, Operation Research, Sociology, Psychology etc  We name a probability with a number from 0 to 1.  If an event is certain to happen, then the probability of the event is 1.  If an event is certain not to happen, then the probability of the event is 0. 1

If it is uncertain whether or not an event will happen, then its probability is some fraction between 0 and 1 (or a fraction converted to a decimal number). Random Experiment An experiment which produces different results even though it is repeated a large number of times under essentially similar conditions is called a random experiment Trial A single performance of an experiment is called a trial Output The results obtained from an experiment is called Output Sample Space The total possible outcome from a random experiment S={HH,HT,TH,TT}S={HHH,HHT,HTH,THH,HTT,THT,TTH,TTT} 2

Sample Point: An individual outcome of an experiment S={HH}Event An event is an individual outcome or any number of outcomes ( sample point) of a random experiment or a trial Single Event Compound Event Mutually exclusive Event/Disjoint Event Two events A and B are said to be mutually exclusive or disjoint if and only if they cannot both occur at the same time Not Mutually Exclusive Event If two events occur at the same time Collectively Exhaustive Event When the union of mutually exclusive events is the entire sample space S Equally likely Event One event is as likely to occur as the other 3

Counting of Sample Points  Rule of Multiplication mxn  Rule of Permutation nPr=n!/(n-r)!  Rule of Combination nCr=n!/(n-r)!r! Question: Determine the number of ways in which a man can wear a suit and a tie if he has three suits and five ties? Page -219 Question: A room has three doors marked A,B,C in how many ways can a person enter by one door but leave by a different door? Question: A system has ten switches, each of which may be either open or closed. The state of the system is described by indicating for each switch whether it is open or closed, how many different states of the system are there? 4

Question : In how many ways can President, VP, Secretary of an executive can be selected from nine members of a committee? Question: How many three-digit numbers can be formed from the digits 1,3,5,7 and 9 if each digit can be used only once? Question: How many possible permutations can be formed from the three letters A, B and C? Question: In how many ways a five-man basket-ball team can be selected from seven men? Question: A bag contains 6,red balls and 4 white balls.Two balls are drawn at random from the bag, Find the probability that 1. both are red 2. one is red and one is white Questions: A box contains 14 identical balls of which 6 are white, 5 red and 3 black Four balls are drawn. What is the probability 1. two are red 2. one is white 5

Assumptions of Probability  For any event 0≥P(A)≤1  For any sure event P(S)=1  If A and B are ME events then P(AUB)=P(A)+P(B) Probability of an Event P(A)= Number of sample points in A Number of sample points in S Number of sample points in S Laws of Probability P(A)= 1-P(A’) Question: A coin is tossed 4 times in succession. What is the probability that at least one head is occur SS=2^4=16P(A’)=15/16P(A)=? 6

Laws of Probability Addition Law If A and B are two events which are not ME then P(AUB)=P(A)+P(B)-P(AUB) Question: If one card is selected at random from a deck of 52 playing cards, what is the probability that the card is a club or a face card or both? Question: A pair of dice are thrown. Find the probability of ge tting a total of either 5 or 11? Question: An investor thinks the probability that stock P will rise tomorrow is 0.70 and that Q will rise is He thinks there is a chance that both will rise. What is his probability that P or Q will rise? 7

Laws of Probability Conditional Probability The probability that B will occur, given that A has occurred, is the probability of AB divided by the probability of A P(A/B)=Number of Sample Points in AпB Number of Sample points in B Number of Sample points in B P(A/B)=n(AпB)/n(B) Dividing N and D by n(S) P(A/B)=P(AпB)/P(B) Question: Two coins are tossed, What is the conditional probability that two heads result, given that there is at least one head? Question: A man tosses two fair dice, What is the conditional probability that the sum of the two dice will be 7, given that (1) The sum is odd (2) the sum is greater than 6. (3) the two dice had the same outcome? 8

Laws of Probability Multiplicative Law/Joint Probability Rule P(AпB)=P(A)P(B/A) = P(B)P(A/B) = P(B)P(A/B) Question: A box contains 15 items, 4 of which are defective and 11 are good. Two items are selected. What is the probability that the first is good and the second is defective ? P(A)=11/15 P(B/A)= 4/14 Question: Box A contains 5 green and 7 red balls. Box B contains 3 green 3 red and 6 yellow balls. A box is selected at random and a ball is drawn at random from it. What is the probability that the ball drawn is green? Box A is selected and a green ball is drawn AandE Box A is selected and a green ball is drawn AandE Box B isEselected and a green ball is drawn BandE Box B isEselected and a green ball is drawn BandE P(E)= P(AпE)+P(BпE) P(A)=1/2 P(E/A)= 5/12 P(E/B)= 3/12 P(E)= P(A) P(E/A)+P(B)P(E/B) 9

Independent and Dependent Events Two events A and B are defined to be Statistically independent if the probability of that one event occurs is not affected by whether the other event has or has not occurred P(A/B)=P(A)P(B/A)=P(B) Then P(AпB)=P(A).P(B) The above formula can be expanded. If A, B, C,..., Z are independent events, then: P(A and B and C and... and Z) = P(A). P(B). P(C)... P(Z) Otherwise DEPENDENT Two ME events are said to be independent if only if P(A)P(B)=0 will be if either P(A)=0 or P(B)=0 If both A and B have non zero value then two events that are independent never be ME 10

Question: Two fair dice, one red and one green, are thrown. Let A denote the event that the red die shows an even number and B the event that the green die shows a 5 or 6. Show that the event A and B are Independent. A= shows that red die has an even numbers B= shows that green die has a 5 or 6 AпB= event that red die shows an even number and green die shows a 5 or 6 P(A)= 18/36 = 1/2 P(B)=12/36 = 1/3 P(AпB)=6/36 = 1/6 For Independence P(AпB)=P(A).P(B) =1/2x1/3=1/6 Hence proved that A & B are independent events 2 nd Method P(A/B)=P(AпB)/P(B)=1/6 = ½= P(A) 1/3 1/3 P(B/A)= P(AпB)/P(A)= 1/6 = 1/3= P(B) 1/2 1/2 11

Question: Find the probability throwing tow consecutive total of 7 in two throws of the die A is the event which shows total 7 A= (1,6), (2,5), (3,4), (4,3), (5,2),(6,1) P(A)= 6/36 B is the event which shows total 7 Same as above P(AпB)=P(A).P(B) P(AпB)=1/6X1/6=1/36 12

Byes Theorem  13

 Question: In bolt factory, machines A,B and C manufacturing 25%,35%,and 40% of the total output, respectively of their outputs, 5, 4 and 2 percent, respectively, are defective bolts. A bolt is selected at random and found to be defective, What is the probability that the bolt came from machine A B C?  P(A)=0.25 P(B)= 0.35 P (C )= 0.40  E represent the event that a bolt is defective  Conditional Probabilities are P( E/A)= 0.05 P(E/B)= 0.04 P(E/C)=0.02 P( E/A)= 0.05 P(E/B)= 0.04 P(E/C)=0.02 P(A/E)= P(A) P(E/A). P(A) P(E/A)+P(B)P(E/B)+P(C) P(E/C) P(A) P(E/A)+P(B)P(E/B)+P(C) P(E/C) = (0.25) (0.05). = = (0.25) (0.05). = (0.25) (0.05)+ (0.35)(0.04)+ (0.40) (0.02) (0.25) (0.05)+ (0.35)(0.04)+ (0.40) (0.02) = = P(B/E)= P(B) P(E/B) = (.35) (0.04) = P(C/E)=???????????????????????? 14

 Question: A box contains 3 white and 2 red marbles while another box contains 2 white and 5 red balls. A marble is selected at random from one of the boxes turns out to be white. What is the probability that it came from the first box?  A and B denote the events that the first and second boxes are chosen respectively.  W denote the event of drawing a white marble P( A)= ½ P(B)=1/2 P(W/A)= 3/5, P(W/B)= 2/7 P( A)= ½ P(B)=1/2 P(W/A)= 3/5, P(W/B)= 2/7 P(A/W)= P(A) P(W/A). P(A) P(W/A)+P(B) P(W/B) P(A) P(W/A)+P(B) P(W/B) P(A/W) = 1/2 x3/5. 1/2x3/5+ 1/2x2/7 1/2x3/5+ 1/2x2/7 = 21/31 = 21/31 15

Question: Bowl I has 2 white and 3 black balls, Bowl II has 4 white and 1 black ball; and Bowl II has 3 white and 4 black balls. A Bowl is selected at random and a ball drawn at random is found to be white. Find the probability that bowl I was selected. Also find probability that Bowl II and Bowl III were selected.????? ( Assignment) 16

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