Probability Probability • A quantitative measure of uncertainty A quantitative measure of uncertainty • A measure of degree of belief in a particular statement A measure of degree of belief in a particular statement or problem or problem • Probability is a measure of how likely it is for an Probability is a measure of how likely it is for an event to happen. event to happen. • The probability and statistics are interrelated The probability and statistics are interrelated • Foundation of Probability were laid by two French Foundation of Probability were laid by two French Mathematician , Blaise Pascal, Pierre De Fermat Mathematician , Blaise Pascal, Pierre De Fermat • Probability theory has a wider field of application and Probability theory has a wider field of application and is used to make intelligent decision in Management , is used to make intelligent decision in Management , Economics, Operation Research, Sociology, Psychology etc Economics, Operation Research, Sociology, Psychology etc We name a probability with a number from 0 to 1. We name a probability with a number from 0 to 1. If an event is certain to happen, then the probability If an event is certain to happen, then the probability of the event is 1. of the event is 1. If an event is certain not to happen, then the If an event is certain not to happen, then the probability of the event is 0. probability of the event is 0. 1
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ProbabilityProbability• A quantitative measure of uncertaintyA quantitative measure of uncertainty• A measure of degree of belief in a particular statement or A measure of degree of belief in a particular statement or
problemproblem• Probability is a measure of how likely it is for an event to Probability is a measure of how likely it is for an event to
happen.happen.• The probability and statistics are interrelated The probability and statistics are interrelated • Foundation of Probability were laid by two French Foundation of Probability were laid by two French
Mathematician , Blaise Pascal, Pierre De FermatMathematician , Blaise Pascal, Pierre De Fermat• Probability theory has a wider field of application and is used Probability theory has a wider field of application and is used
to make intelligent decision in Management , Economics, to make intelligent decision in Management , Economics, Operation Research, Sociology, Psychology etcOperation Research, Sociology, Psychology etc
We name a probability with a number from 0 to 1. We name a probability with a number from 0 to 1. If an event is certain to happen, then the probability of the If an event is certain to happen, then the probability of the
event is 1.event is 1. If an event is certain not to happen, then the probability of If an event is certain not to happen, then the probability of
the event is 0.the event is 0.
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If it is uncertain whether or not an event will happen, then its If it is uncertain whether or not an event will happen, then its probability is some fraction between 0 and 1 (or a fraction probability is some fraction between 0 and 1 (or a fraction converted to a decimal number).converted to a decimal number).
Random ExperimentRandom ExperimentAn experiment which produces different results even though it An experiment which produces different results even though it is repeated a large number of times under essentially similar is repeated a large number of times under essentially similar conditions is called a random experimentconditions is called a random experiment
TrialTrialA single performance of an experiment is called a trialA single performance of an experiment is called a trial
OutputOutputThe results obtained from an experiment is called OutputThe results obtained from an experiment is called Output
Sample SpaceSample SpaceThe total possible outcome from a random experimentThe total possible outcome from a random experiment
An individual outcome of an experimentAn individual outcome of an experiment S={HH}S={HH}
Event Event
An event is an individual outcome or any number of outcomes An event is an individual outcome or any number of outcomes ( sample point) of a random experiment or a trial( sample point) of a random experiment or a trial
Two events A and B are said to be mutually exclusive or Two events A and B are said to be mutually exclusive or disjoint if and only if they cannot both occur at the same timedisjoint if and only if they cannot both occur at the same time
Not Mutually Exclusive Event Not Mutually Exclusive Event
If two events occur at the same time If two events occur at the same time
When the union of mutually exclusive events is the entire When the union of mutually exclusive events is the entire sample space Ssample space S
Equally likely EventEqually likely Event
One event is as likely to occur as the other One event is as likely to occur as the other 33
Counting of Sample PointsCounting of Sample Points Rule of Multiplication mxnRule of Multiplication mxn Rule of Permutation nPr=n!/(n-r)!Rule of Permutation nPr=n!/(n-r)! Rule of Combination nCr=n!/(n-r)!r!Rule of Combination nCr=n!/(n-r)!r!QuestionQuestion: Determine the number of ways in which a man can : Determine the number of ways in which a man can
wear a suit and a tie if he has three suits and five ties?wear a suit and a tie if he has three suits and five ties?
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QuestionQuestion: A room has three doors marked A,B,C in how many : A room has three doors marked A,B,C in how many ways can a person enter by one door but leave by a ways can a person enter by one door but leave by a different door?different door?
QuestionQuestion: A system has ten switches, each of which may be : A system has ten switches, each of which may be either open or closed. The state of the system is described either open or closed. The state of the system is described by indicating for each switch whether it is open or closed , by indicating for each switch whether it is open or closed , how many different states of the system are there?how many different states of the system are there?
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QuestionQuestion: : In how many ways can President , VP, Secretary of In how many ways can President , VP, Secretary of an executive can be selected from nine members of a an executive can be selected from nine members of a committee?committee?
QuestionQuestion: How many three-digit numbers can be formed from : 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?the digits 1,3,5,7 and 9 if each digit can be used only once?
QuestionQuestion: How many possible permutations can be formed from : How many possible permutations can be formed from the three letters A, B and C?the three letters A, B and C?
QuestionQuestion: In how many ways a five-man basket-ball team can : In how many ways a five-man basket-ball team can be selected from seven men?be selected from seven men?
QuestionQuestion: A bag contains 6 ,red balls and 4 white balls.Two : A bag contains 6 ,red balls and 4 white balls.Two balls are drawn at random from the bag, Find the probability balls are drawn at random from the bag, Find the probability that 1. both are red 2. one is red and one is white that 1. both are red 2. one is red and one is white
QuestionsQuestions: A box contains 14 identical balls of which 6 are : A box contains 14 identical balls of which 6 are white, 5 red and 3 black Four balls are drawn. What is the white, 5 red and 3 black Four balls are drawn. What is the probability 1. two are red 2. one is whiteprobability 1. two are red 2. one is white
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Assumptions of ProbabilityAssumptions of Probability For any event 0≥P(A)≤1For any event 0≥P(A)≤1 For any sure event P(S)=1 For any sure event P(S)=1 If A and B are ME events then P(AUB)=P(A)+P(B)If A and B are ME events then P(AUB)=P(A)+P(B)
Probability of an EventProbability of an Event
P(A)= P(A)= Number of sample points in ANumber of sample points in A
Number of sample points in SNumber of sample points in S
Laws of ProbabilityLaws of Probability
P(A)= 1-P(A’)P(A)= 1-P(A’)Question: Question: A coin is tossed 4 times in succession. What is the A coin is tossed 4 times in succession. What is the
probability that at least one head is occurprobability that at least one head is occur
SS=2^4=16SS=2^4=16
P(A’)=15/16P(A’)=15/16
P(A)=?P(A)=?
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Laws of ProbabilityLaws of ProbabilityAddition LawAddition LawIf A and B are two events which are not ME then P(AUB)=P(A)If A and B are two events which are not ME then P(AUB)=P(A)+P(B)-P(AUB)+P(B)-P(AUB)
QuestionQuestion: If one card is selected at random from a deck of 52 : 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 playing cards, what is the probability that the card is a club or a face card or both?a face card or both?
QuestionQuestion: A pair of dice are thrown. Find the probability of ge : A pair of dice are thrown. Find the probability of ge tting a total of either 5 or 11?tting a total of either 5 or 11?
QuestionQuestion: An investor thinks the probability that stock P will : An investor thinks the probability that stock P will rise tomorrow is 0.70 and that Q will rise is 0.80. He thinks rise tomorrow is 0.70 and that Q will rise is 0.80. He thinks there is a 50-50 chance that both will rise. What is his there is a 50-50 chance that both will rise. What is his probability that P or Q will rise?probability that P or Q will rise?
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Laws of ProbabilityLaws of Probability Conditional ProbabilityConditional Probability The probability that B will occur, given that A has occurred , is The probability that B will occur, given that A has occurred , is the probability of AB divided by the probability of Athe probability of AB divided by the probability of A
P(A/B)=P(A/B)=Number of Sample Points in ANumber of Sample Points in AппBB
Number of Sample points in BNumber of Sample points in B
P(A/B)=n(AP(A/B)=n(AппB)/n(B) Dividing N and D by n(S)B)/n(B) Dividing N and D by n(S)
P(A/B)=P(AP(A/B)=P(AппB)/P(B)B)/P(B)
QuestionQuestion: Two coins are tossed, What is the conditional : Two coins are tossed, What is the conditional probability that two heads result, given that there is at least probability that two heads result, given that there is at least one head?one head?
Question:Question: A man tosses two fair dice, What is the conditional A man tosses two fair dice, What is the conditional probability that the sum of the two dice will be 7 , given that 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 (1) The sum is odd (2) the sum is greater than 6. (3) the two dice had the same outcome?dice had the same outcome?
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Laws of Probability Laws of Probability Multiplicative Law/Joint Probability RuleMultiplicative Law/Joint Probability Rule
P(AP(AппB)=P(A)P(B/A)B)=P(A)P(B/A)
= P(B)P(A/B)= P(B)P(A/B)Question: Question: A box contains 15 items , 4 of which are defective A box contains 15 items , 4 of which are defective and 11 are good. Two items are selected. What is the probability and 11 are good. Two items are selected. What is the probability that the first is good and the second is defective ?that the first is good and the second is defective ?
P(A)=11/15 P(B/A)= 4/14P(A)=11/15 P(B/A)= 4/14
Question: Box A contains 5 green and 7 red balls. Box B contains 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 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 a ball is drawn at random from it. What is the probability that the ball drawn is green?ball drawn is green?
Box A is selected and a green ball is drawn AandEBox A is selected and a green ball is drawn AandE
Box B isEselected and a green ball is drawn BandEBox B isEselected and a green ball is drawn BandE
Independent and Dependent EventsIndependent and Dependent EventsTwo events A and B are defined to be Statistically Two events A and B are defined to be Statistically independent if the probability of that one event occurs independent if the probability of that one event occurs is not affected by whether the other event has or has is not affected by whether the other event has or has not occurrednot occurred
P(A/B)=P(A) P(A/B)=P(A)
P(B/A)=P(B)P(B/A)=P(B)
Then P(AThen P(AппB)=P(A).P(B)B)=P(A).P(B)The above formula can be expanded. If A , B , C , ... , Z are The above formula can be expanded. If A , B , C , ... , Z are independent events, then:independent events, then:
P(A and B and C and ... and Z) = P(A) . P(B) . P(C) ... P(Z)P(A and B and C and ... and Z) = P(A) . P(B) . P(C) ... P(Z)
Otherwise DEPENDENTOtherwise DEPENDENT
Two ME events are said to be independent if only if 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)=0P(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 If both A and B have non zero value then two events that are independent never be ME that are independent never be ME 1010
Question:Question: Two fair dice, one red and one green, are thrown. Let A 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 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 event that the green die shows a 5 or 6. Show that the event A and B are Independent.are Independent.
A= shows that red die has an even numbersA= shows that red die has an even numbers
B= shows that green die has a 5 or 6B= shows that green die has a 5 or 6
AAппB= event that red die shows an even number and green die shows B= event that red die shows an even number and green die shows a 5 or 6a 5 or 6
P(A)= 18/36 = 1/2P(A)= 18/36 = 1/2
P(B)=12/36 = 1/3P(B)=12/36 = 1/3
P(AP(AппB)=6/36 = 1/6B)=6/36 = 1/6
For Independence P(AFor Independence P(AппB)=P(A).P(B) =1/2x1/3=1/6B)=P(A).P(B) =1/2x1/3=1/6
Hence proved that A & B are independent eventsHence proved that A & B are independent events
Question:Question: Find the probability throwing tow consecutive total of Find the probability throwing tow consecutive total of 7 in two throws of the die7 in two throws of the die
A is the event which shows total 7A is the event which shows total 7
B is the event which shows total 7B is the event which shows total 7
Same as above Same as above
P(AP(AппB)=P(A).P(B)B)=P(A).P(B)
P(AP(AппB)=1/6X1/6=1/36B)=1/6X1/6=1/36
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Byes TheoremByes Theorem
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Question: Question: In bolt factory, machines A,B and C manufacturing In bolt factory, machines A,B and C manufacturing 25%,35%,and 40% of the total output, respectively of their 25%,35%,and 40% of the total output, respectively of their outputs , 5, 4 and 2 percent, respectively, are defective bolts. A outputs , 5, 4 and 2 percent, respectively, are defective bolts. A bolt is selected at random and found to be defective , What is the bolt is selected at random and found to be defective , What is the probability that the bolt came from machine A B C? probability that the bolt came from machine A B C?
P(A)=0.25 P(B)= 0.35 P (C )= 0.40 P(A)=0.25 P(B)= 0.35 P (C )= 0.40
E represent the event that a bolt is defective E represent the event that a bolt is defective Conditional Probabilities are Conditional Probabilities are
Question:Question: A box contains 3 white and 2 red marbles while A box contains 3 white and 2 red marbles while another box contains 2 white and 5 red balls. A marble is another box contains 2 white and 5 red balls. A marble is selected at random from one of the boxes turns out to be selected at random from one of the boxes turns out to be white. What is the probability that it came from the first white. What is the probability that it came from the first box?box?
A and B denote the events that the first and second boxes A and B denote the events that the first and second boxes are chosen respectively.are chosen respectively.
W denote the event of drawing a white marbleW denote the event of drawing a white marble
P( A)= ½ P(B)=1/2 P(W/A)= 3/5, P(W/B)= 2/7P( A)= ½ P(B)=1/2 P(W/A)= 3/5, P(W/B)= 2/7
P(A/W)= P(A/W)= P(A) P(W/A) . 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) = P(A/W) = 1/2 x3/5 . 1/2 x3/5 .
1/2x3/5+ 1/2x2/71/2x3/5+ 1/2x2/7
= 21/31= 21/31
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Question: Bowl I has 2 white and 3 black balls, Bowl Question: Bowl I has 2 white and 3 black balls, Bowl II has 4 white and 1 black ball; and Bowl II has 3 II has 4 white and 1 black ball; and Bowl II has 3 white and 4 black balls . A Bowl is selected at white and 4 black balls . A Bowl is selected at random and a ball drawn at random is found to be random and a ball drawn at random is found to be white. Find the probability that bowl I was selected . white. Find the probability that bowl I was selected . Also find probability that Bowl II and Bowl III were Also find probability that Bowl II and Bowl III were selected.????? ( Assignment)selected.????? ( Assignment)
