Aptitude Round QuestionProbabilityFree Download
An operation which can produce some welldefined outcomes is called an
experiment.
An experiment in which all possible outcomes are know and the exact output
cannot be predicted in advance, is called a random experiment.
Examples:

Rolling an unbiased dice.

Tossing a fair coin.

Drawing a card from a pack of wellshuffled cards.

Picking up a ball of certain colour from a bag containing balls of
different colours.
Details:

When we throw a coin, then either a Head (H) or a Tail (T) appears.

A dice is a solid cube, having 6 faces, marked 1, 2, 3, 4, 5, 6
respectively. When we throw a die, the outcome is the number that
appears on its upper face.

A pack of cards has 52 cards.
It has 13 cards of each suit, name Spades, Clubs, Hearts and Diamonds.
Cards of spades and clubs are black cards.
Cards of hearts and diamonds are red cards.
There are 4 honours of each unit.
There are Kings, Queens and Jacks. These are all called face cards.
When we perform an experiment, then the set S of all possible outcomes is
called the sample space.
Examples:

In tossing a coin, S = {H, T}

If two coins are tossed, the S = {HH, HT, TH, TT}.

In rolling a dice, we have, S = {1, 2, 3, 4, 5, 6}.
Any subset of a sample space is called an event.

Probability of Occurrence of an Event:
Let S be the sample and let E be an event.
Then, E S.

P(S) = 1

0 P (E) 1

P() = 0

For any events A and B we have : P(A B) = P(A) + P(B)  P(A B)

If A denotes (notA), then P(A) = 1  P(A).
Important Formulas  Probability or Chance
Probability or chance is a common term used in daytoday life. For
example, we generally say, 'it may rain today'. This statement has a
certain uncertainty.
Probability is quantitative measure of the chance of occurrence of a
particular event.
An experiment is an operation which can produce welldefined outcomes.
If all the possible outcomes of an experiment are known but the exact
output cannot be predicted in advance, that experiment is called a random
experiment.
Examples

Tossing of a fair coin
When we toss a coin, the outcome will be either Head (H) or Tail (T)

Throwing an unbiased die
Die is a small cube used in games. It has six faces and each of the six
faces shows a different number of dots from 1 to 6. Plural of die is dice.
When a die is thrown or rolled, the outcome is the number that appears on
its upper face and it is a random integer from one to six, each value being
equally likely.

Drawing a card from a pack of shuffled cards
A pack or deck of playing cards has 52 cards which are divided into four
categories as given below
a. Spades (?)
b. Clubs (?)
c. Hearts (?)

Diamonds (?)
Each of the above mentioned categories has 13 cards, 9 cards numbered from
2 to 10, an Ace, a King, a Queen and a jack
Hearts and Diamonds are red faced cards whereas Spades and Clubs are black
faced cards.
Kings, Queens and Jacks are called face cards

Taking a ball randomly from a bag containing balls of different
colours
Sample Space
Sample Space is the set of all possible outcomes of an experiment. It is
denoted by S.
Examples

When a coin is tossed, S = {H, T} where H = Head and T = Tail

When a dice is thrown, S = {1, 2 , 3, 4, 5, 6}

When two coins are tossed, S = {HH, HT, TH, TT} where H = Head and T =
Tail
Event
Any subset of a Sample Space is an event. Events are generally denoted by
capital letters A, B , C, D etc.
Examples

When a coin is tossed, outcome of getting head or tail is an event

When a die is rolled, outcome of getting 1 or 2 or 3 or 4 or 5 or 6 is
an event
Equally Likely Events
Events are said to be equally likely if there is no preference for a
particular event over the other.
Examples

When a coin is tossed, Head (H) or Tail is equally likely to occur.

When a dice is thrown, all the six faces (1, 2, 3, 4, 5, 6) are equally
likely to occur.
Mutually Exclusive Events
Two or more than two events are said to be mutually exclusive if the
occurrence of one of the events excludes the occurrence of the other
This can be better illustrated with the following examples

When a coin is tossed, we get either Head or Tail. Head and Tail cannot
come simultaneously. Hence occurrence of Head and Tail are mutually
exclusive events.

When a die is rolled, we get 1 or 2 or 3 or 4 or 5 or 6. All these
faces cannot come simultaneously. Hence occurrences of particular faces
when rolling a die are mutually exclusive events.
Note : If A and B are mutually exclusive events, A $MF#%\cap$MF#% B =
$MF#%\phi$MF#% where $MF#%\phi$MF#% represents empty set.

Consider a die is thrown and A be the event of getting 2 or 4 or 6 and
B be the event of getting 4 or 5 or 6. Then
A = {2, 4, 6} and B = {4, 5, 6}
Here A $MF#%\cap$MF#% B $MF#% \neq \phi$MF#%. Hence A and B are not
mutually exclusive events.
Events can be said to be independent if the occurrence or nonoccurrence of
one event does not influence the occurrence or nonoccurrence of the other.
Example
: When a coin is tossed twice, the event of getting Tail(T) in the first
toss and the event of getting Tail(T) in the second toss are independent
events. This is because the occurrence of getting Tail(T) in any toss does
not influence the occurrence of getting Tail(T) in the other toss.
In the case of simple events, we take the probability of occurrence of
single events.
Examples

Probability of getting a Head (H) when a coin is tossed

Probability of getting 1 when a die is thrown
Compound Events
In the case of compound events, we take the probability of joint occurrence
of two or more events.
Examples

When two coins are tossed, probability of getting a Head (H) in the
first toss and getting a Tail (T) in the second toss.
Exhaustive Events
Exhaustive Event is the total number of all possible outcomes of an
experiment.
Examples

When a coin is tossed, we get either Head or Tail. Hence there are 2
exhaustive events.

When two coins are tossed, the possible outcomes are (H, H), (H, T),
(T, H), (T, T). Hence there are 4 (=2^{2}) exhaustive events.

When a dice is thrown, we get 1 or 2 or 3 or 4 or 5 or 6. Hence there
are 6 exhaustive events.
Algebra of Events
Let A and B are two events with sample space S. Then

A $MF#%\cup$MF#% B is the event that either A or B or Both occur.
(i.e., at least one of A or B occurs)

A $MF#%\cap$MF#% B is the event that both A and B occur

$MF#%\bar{\text{A}}$MF#% is the event that A does not occur

$MF#%\bar{\text{A}}\cap\bar{\text{B}} $MF#% is the event that none of A
and B occurs
Example
: Consider a die is thrown , A be the event of getting 2 or 4 or 6 and B be
the event of getting 4 or 5 or 6. Then
A = {2, 4, 6} and B = {4, 5, 6}
A $MF#%\cup$MF#% B = {2, 4, 5, 6}
A $MF#%\cap$MF#% B = {4, 6}
$MF#%\bar{\text{A}}$MF#% = {1, 3, 5}
$MF#%\bar{\text{B}}$MF#% = {1, 2, 3}
$MF#%\bar{\text{A}}\cap\bar{\text{B}} $MF#% = {1,3}
Let E be an event and S be the sample space. Then probability of the event
E can be defined as
P(E) = $MF#%\dfrac{\text{n(E)}}{\text{n(S)}}$MF#%
where P(E) = Probability of the event E, n(E) = number of ways in which the
event can occur and n(S) = Total number of outcomes possible
Examples

A coin is tossed once. What is the probability of getting Head?
Total number of outcomes possible when a coin is tossed = n(S) = 2 (? Head
or Tail)
E = event of getting Head = {H}. Hence n(E) = 1
$MF#%\text{P(E) = }\dfrac{\text{n(E)}}{\text{n(S)}} = \dfrac{1}{2}$MF#%

Two dice are rolled. What is the probability that the sum on the top
face of both the dice will be greater than 9?
Total number of outcomes possible when a die is rolled = 6 (? any one face
out of the 6 faces)
Hence, total number of outcomes possible two dice are rolled, n(S) = 6 × 6
= 36
E = Getting a sum greater than 9 when the two dice are rolled = {(4, 6),
{5, 5}, {5, 6}, {6, 4}, {6, 5}, (6, 6)}
Hence, n(E) = 6
$MF#%\text{P(E) = }\dfrac{\text{n(E)}}{\text{n(S)}} = \dfrac{6}{36} =
\dfrac{1}{6}$MF#%
P(S) = 1
0 $MF#%\leq$MF#% P (E) $MF#%\leq$MF#% 1
P($MF#%\phi$MF#%) = 0 (? Probability of occurrence of an impossible event =
0)
Let A and B be two events associated with a random experiment. Then
P(A U B) = P(A) + P(B) – P(A $MF#%\cap$MF#% B)
If A and B are mutually exclusive events, then P(A U B) = P(A) + P(B)
because for mutually exclusive events, P(A $MF#%\cap$MF#% B) = 0
P(A $MF#%\cap$MF#% B) = P(A).P(B)
Example
: Two dice are rolled. What is the probability of getting an odd number in
one die and getting an even number in the other die?
Total number of outcomes possible when a die is rolled, n(S) = 6 (? any one
face out of the 6 faces)
Let A be the event of getting the odd number in one die = {1,3,5}. =>
n(A)= 3
$MF#%\text{P(A) = }\dfrac{\text{n(A)}}{\text{n(S)}} = \dfrac{3}{6} =
\dfrac{1}{2}$MF#%
Let B be the event of getting an even number in the other die = {2,4, 6}.
=> n(B)= 3
$MF#%\text{P(B) = }\dfrac{\text{n(B)}}{\text{n(S)}} = \dfrac{3}{6} =
\dfrac{1}{2}$MF#%
Required Probability, P(A $MF#%\cap$MF#% B) = P(A).P(B) = $MF#%\dfrac{1}{2}
\times \dfrac{1}{2} = \dfrac{1}{4}$MF#%
Let A be any event and $MF#%\bar{\text{A}}$MF#% be its complementary event
(i.e., $MF#%\bar{\text{A}}$MF#% is the event that A does not occur). Then
$MF#%\text{P(}\bar{\text{A}}\text{)}$MF#% = 1  P(A)
Let E be an event associated with a random experiment. Let $MF#%x$MF#%
outcomes are favourable to E and y outcomes are not favourable to E, then
Odds in favour of E are $MF#%x : y$MF#%, i.e., $MF#%\dfrac{x}{y}$MF#% and
Odds against E are $MF#%y : x$MF#%, i.e., $MF#%\dfrac{y}{x}$MF#%
P(E) = $MF#%\dfrac{x}{x+y}$MF#%
$MF#%\text{P(}\bar{\text{E}}\text{)}$MF#% =$MF#%\dfrac{y}{x+y}$MF#%
Example
: What are the odds in favour of and against getting a 1 when a die is
rolled?
Let E be an event of getting 1 when a die is rolled
Outcomes which are favourable to E, $MF#%x = 1$MF#%
Outcomes which are not favourable to E, $MF#%y = 5$MF#%
Odds in favour of getting 1 = $MF#%\dfrac{x}{y} = \dfrac{1}{5}$MF#%
Odds against getting 1 = $MF#%\dfrac{x}{y} = \dfrac{y}{x} =
\dfrac{5}{1}$MF#%
Let A and B be two events associated with a random experiment. Then,
probability of the occurrence of A given that B has already occurred is
called conditional probability and denoted by P(A/B)
Example : A bag contains 5 black and 4 blue balls. Two balls are drawn from
the bag one by one without replacement. What is the probability of drawing
a blue ball in the second draw if a black ball is already drawn in the
first draw?
Let A be the event of drawing black ball in the first draw and B be the
event of drawing a blue ball in the second draw. Then, P(B/A) = Probability
of drawing a blue ball in the second draw given that a black ball is
already drawn in the first draw.
Total Balls = 5 + 4 = 9
Since a black ball is drawn already,
total number of balls left after the first draw = 8
total number of blue balls after the first draw = 4
$MF#%\text{P(B/A) }= \dfrac{4}{8} = \dfrac{1}{2}$MF#%
A binomial experiment is a probability experiment which satisfies the
following requirements.
1. Each trial can have only two outcomes. These outcomes can be considered
as either success or failure.
2. There must be a fixed number of trials.
3. The outcomes of each trial must be independent of each other.
4. The probability of a success must remain the same for each trial.

Tickets numbered 1 to 20 are mixed up and then a ticket is
drawn at random. What is the probability that the ticket
drawn has a number which is a multiple of 3 or 5?

Answer: Option D


A bag contains 2 red, 3 green and 2 blue balls. Two balls
are drawn at random. What is the probability that none of
the balls drawn is blue?

Answer: Option A


In a box, there are 8 red, 7 blue and 6 green balls. One
ball is picked up randomly. What is the probability that it
is neither red nor green?

Answer: Option A


What is the probability of getting a sum 9 from two throws
of a dice?

Answer: Option C


Three unbiased coins are tossed. What is the probability of
getting at most two heads?

Answer: Option D

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