Transcript probability - The Toppers Way

“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)
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
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.
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
nC
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
A1
B
A2
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)
Computation of Posterior Probabilities
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)
P(B)
1.00
A1
A2
.
.
.
Total
1.00
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
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.