No. 
Problems & Solutions 
01. 
In a tossing a die, the total number of possible out comes =
(Or) In a random experiment of rolling a die, what are the elementary events.
 
In the experiment of tossing an unbiased dice/die, there are six possible elementary events:
The events of the dice/die showing up the number
"ONE" "TWO" "THREE" "FOUR" "FIVE" (Or) "SIX"
⇒ The total number of possible choices in the experiment = 6

No. 
Problems & Solutions 
02. 
When a cubical die is rolled, find the probability of getting an even integer. Find also the odds for the event.
 
In the experiment of rolling a cubical dice/die,
Total No. of Possible Choices 
= 
6 {ONE, TWO, THREE, FOUR, FIVE, SIX} 
⇒ n 
= 
6 
Let "A" be the event of getting an even integer.
For Event "A"
No. of Favorable Choices 
= 
3 {TWO, FOUR, SIX} 
⇒ m_{A} 
= 
3 
Probability of getting an even integer
⇒ Probability of occurrence of Event "A" 
= 
Number of Favourable/Favorable Choices for the Event  Total Number of Possible Choices for the Experiment 

⇒ P(A) 
= 


= 


= 

• Odds
No. of UnFavorable Choices

= 
Total No. of possible choices − No. of Favorable choices 
⇒ m_{A}^{c} 
= 
n − m_{A}


= 
6 − 3 

= 
3 
» in favor/favour
Odds in Favour of getting an even integer
⇒ Odds in Favor/Favour of Event "A" 
= 
No. of Favourable Choices : No. of UnFavorable Choices 

= 
m : m_{A}^{c} 

= 
3 : 3 

= 
1 : 1 
» against
Odds against getting an even integer
⇒ Odds against Event "A" 
= 
No. of UnFavourable Choices : No. of Favorable Choices 

= 
m_{A}^{c} : m 

= 
3 : 3 

= 
1 : 1 

No. 
Problems & Solutions 
03. 
Find the probability and odds in favour/favor of getting an odd number or a multiple of 4 on throwing a dice
 
In the experiment of throwing a dice/die,
Total No. of Possible Choices 
= 
6 {ONE, TWO, THREE, FOUR, FIVE, SIX} 
⇒ n 
= 
6 
Let "A" be the event of getting an odd number or a multiple of 4.
For Event "A"
No. of Favorable Choices 
= 
4 {ONE, THREE, FOUR, FIVE} 
⇒ m_{A} 
= 
4 
Probability of getting an even integer
⇒ Probability of occurrence of Event "A" 
= 
Number of Favourable/Favorable Choices for the Event  Total Number of Possible Choices for the Experiment 

⇒ P(A) 
= 


= 


= 

• Odds
No. of UnFavorable Choices

= 
Total No. of possible choices − No. of Favorable choices 
⇒ m_{A}^{c} 
= 
n − m_{A}


= 
6 − 4 

= 
2 
» in favor/favour
Odds in Favour of getting an odd number or a multiple of 4
⇒ Odds in Favor/Favour of Event "A" 
= 
No. of Favourable Choices : No. of UnFavorable Choices 

= 
m : m_{A}^{c} 

= 
4 : 2 

= 
2 : 1 

No. 
Problems & Solutions 
04. 
A (six  faced) die is thrown, find the chance that an even number more than 2 does not turn up?
 
In the experiment of throwing a dice/die,
Total No. of Possible Choices = 6 {ONE, TWO, THREE, FOUR, FIVE, SIX} ⇒ n = 6
Let "A" be the event of an even number more than 2 turning up.
For Event "A"
No. of Favorable Choices 
= 
2 {FOUR, SIX} 
⇒ m_{A} 
= 
2 
Probability that an even number more than 2 turns up
⇒ Probability of occurrence of Event "A" 
= 
Number of Favourable/Favorable Choices for the Event  Total Number of Possible Choices for the Experiment 

⇒ P(A) 
= 


= 


= 

Probability that an even number more than 2 does not turn up
⇒ Probability of nonoccurrence of Event "A"

= 
1 − Probability of occurrence of Event A 
P(A^{c}) 
= 
1 − P(A) 

= 


= 


= 

• Alternative
No. of UnFavorable Choices

= 
Total No. of possible choices − No. of Favorable choices 
⇒ m_{A}^{c} 
= 
n − m_{A}


= 
6 − 2 

= 
4 
Probability that an even number more than 2 does not turn up
⇒ Probability of nonoccurrence of Event "A" 
= 
Number of UnFavourable/UnFavorable Choices for the Event  Total Number of Possible Choices for the Experiment 

⇒ P(A^{c}) 
= 


= 


= 


No. 
Problems & Solutions 
0. 
A die is thrown once, find P (a number ≥ 4) and also the odds.
 
In the experiment of throwing a dice/die once,
Total No. of Possible Choices 
= 
6 {ONE, TWO, THREE, FOUR, FIVE, SIX} 
⇒ n 
= 
6 
Let "A" be the event of getting a number ≥ 4.
For Event "A"
No. of Favorable Choices 
= 
3 {FOUR, FIVE, SIX} 
⇒ m_{A} 
= 
3 
Probability of getting a number ≥ 4
⇒ Probability of occurrence of Event "A" 
= 
Number of Favourable/Favorable Choices for the Event  Total Number of Possible Choices for the Experiment 

⇒ P(A) 
= 


= 


= 

• Odds
No. of UnFavorable Choices

= 
Total No. of possible choices − No. of Favorable choices 
⇒ m_{A}^{c} 
= 
n − m_{A}


= 
6 − 3 

= 
3 
» in favor/favour
Odds in Favour of getting a number ≥ 4
⇒ Odds in Favor/Favour of Event "A" 
= 
No. of Favourable Choices : No. of UnFavorable Choices 

= 
m : m_{A}^{c} 

= 
3 : 3 

= 
1 : 1 
» against
Odds against getting a number ≥ 4
⇒ Odds against Event "A" 
= 
No. of UnFavourable Choices : No. of Favorable Choices 

= 
m_{A}^{c} : m 

= 
3 : 3 

= 
1 : 1 

No. 
Problems & Solutions 
06. 
Define the Event and identify the number of favourable and choices in the following which relate to the experiment of rolling a dice/die:
a) 
Getting 4 when a dice is rolled 
b) 
Getting a face having a number less than 5? 
c) 
Throwing a number greater than 2. 
d) 
The number appearing on top is not an even number 
e) 
Getting 3 and 5 simultaneously 
f) 
Getting 4 or 6 in a throw of single die is 
g) 
An an odd number less than 4 turns up 
h) 
An ace turns up 
i) 
Getting 7 
j) 
Getting an Even number or a multiple of 3 
 
In the experiment of tossing a dice/die
Total No. of Possible Choices 
= 
6 {ONE, TWO, THREE, FOUR, FIVE, SIX} 
⇒ n 
= 
6 
a) 
Let "A" be the event of getting 4 when a die is rolled
For Event "A"
No. of Favorable Choices 
= 
1 {FOUR} 
⇒ m_{A} 
= 
1 

b) 
Let "B" be the event of getting a face having a number less than 5 on tossing a die
For Event "B"
No. of Favorable Choices 
= 
4 {ONE, TWO, THREE, FOUR} 
⇒ m_{B} 
= 
4 

c) 
Let "C" be the event of throwing a number greater than 2
For Event "C"
No. of Favorable Choices 
= 
4 {THREE, FOUR, FIVE, SIX} 
⇒ m_{C} 
= 
4 

d) 
Let "D" be the event of the number appearing on the die not being an even number
For Event "D"
No. of Favorable Choices 
= 
3 {ONE, THREE, FIVE} 
⇒ m_{D} 
= 
3 

e) 
Let "E" be the event of getting 3 and 5 simultaneously
For Event "E"
No. of Favorable Choices 
= 
0 {Φ} 
⇒ m_{E} 
= 
0 
Since only One number can appear on throwing a die, the numbers 3 and 5 cannot appear simultaneously.

f) 
Let "F" be the event of getting 4 or 6 on a single throw
For Event "F"
No. of Favorable Choices 
= 
2 {FOUR, SIX} 
⇒ m_{F} 
= 
2 

g) 
Let "G" be the event of an odd number less than 4 turning up
For Event "G"
No. of Favorable Choices 
= 
2 {ONE, THREE} 
⇒ m_{G} 
= 
2 

h) 
Let "H" be the event that an ACE turns up
For Event "H"
No. of Favorable Choices 
= 
1 {ONE} 
⇒ m_{H} 
= 
1 

i) 
Let "I" be the event of getting 7
For Event "I"
No. of Favorable Choices 
= 
0 {Φ} 
⇒ m_{I} 
= 
0 
Since only numbers from One to Six can appear on throwing a die, the numbers 7 cannot appear on the dice.

j) 
Let "J" be the event of getting an even number or a multiple of 3
For Event "J"
No. of Favorable Choices 
= 
4 {TWO, THREE, FOUR, SIX} 
⇒ m_{J} 
= 
4 


No. 
Problems for Practice 
01. 
What is the chance of throwing a 5 with an ordinary dice?

02. 
The probability of getting an odd number when we throw a single die is

03. 
The probability of getting a number less than four when a die is rolled is __

04. 
Find the probability of throwing a number greater than 4 when a die is rolled

05. 
In a throw of a single die the probability of getting 3 or 5 is ___?

06. 
A dice is rolled, find 
(1) 
P(even number) 
(2) 
P(a number > 1) 
(3) 
P(a number < 5) 
(4) 
P(a number more than 6) 
(5) 
P(a number < 7) 

07. 
When a perfect die is rolled what is the probability of getting a face having 
(1) 
4 Points 
(2) 
Odd Number 
(3) 
2 Points Or 3 Points 

08. 
a) 
Find the probability of getting 2 when a die is rolled 
b) 
What is the probability of throwing a number greater than 3 with an ordinary dice? 
c) 
A die is rolled. What is the probability that a number 1 or 6 may appear on the upper face? 
d) 
A (sixfaced) die is thrown. Find the chance that any one of 1, 2, 3 turns up? 
e) 
If a die is tossed, what is the probability that the number appearing on top is
(i) (ii) (iii) (iv) (v) 
even less than 4 not an even number either an even or an odd number an odd number less than 4. 


09. 
The probability of not getting 1, when a die is rolled

10. 
If a die is tossed, what is the chance of getting an even number greater than 2

11. 
Find the chance of not throwing an ace, two or three in a single throw with a die.

12. 
what are the odds against throwing ace or six in a single throw with a die? And what are the odds in favour?


