Download File

Probability
•
•
•
•
•
•



A quantitative measure of uncertainty
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.
The probability and statistics are interrelated
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
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
3
One event is as likely to occur as the other
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
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=16
P(A’)=15/16
P(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 0.80. He thinks
there is a 50-50 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
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)
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 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
10
that are independent never be ME
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
2nd Method
P(A/B)=P(AпB)/P(B)=1/6 = ½= P(A)
1/3
P(B/A)= P(AпB)/P(A)= 1/6 = 1/3= P(B)
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(A/E)=
=
P(A) P(E/A)
.
P(A) P(E/A)+P(B)P(E/B)+P(C) P(E/C)
(0.25) (0.05)
.
(0.25) (0.05)+ (0.35)(0.04)+ (0.40) (0.02)
= 0.0125
0.0345
= 0.362
P(B/E)= P(B) P(E/B) = (.35) (0.04) = 0.406
0.0345
0.0345
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/W)=
P(A) P(W/A)
.
P(A) P(W/A)+P(B) P(W/B)
P(A/W) =
1/2 x3/5
.
1/2x3/5+ 1/2x2/7
= 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
The End
17