Throwing/Tossing/Rolling Single/One Dice/Die  Probability  Problems & Solutions
Problem 1
(Or)
In a random experiment of rolling a die, what are the elementary events.
Solution
The events of the dice/die showing up the number
ONE TWO THREE FOUR FIVE SIX
⇒ The total number of possible choices in the experiment = 6
Problem 2
Solution
Total Number of Possible Choices
= 6 {ONE, TWO, THREE, FOUR, FIVE, SIX}
⇒ n = 6
Let
A : the event of getting an even integer.
For Event A
= 3 {TWO, FOUR, SIX}
⇒ m_{A} = 3
Probability of getting an an even integer on rolling a dice
⇒ Probability of occurrence of Event A
= 

⇒ P(A)  = 
 
= 
 
= 

Odds
Number of Unfavorable Choices= Total Number of possible choices − Number of Favorable choices
⇒ m_{A}^{c}  =  n − m_{A} 
=  6 − 3  
=  3 
in favor
Odds in Favor of getting an even integer⇒ Odds in Favor of Event A
= Number of Favorable Choices : Number of Unfavorable Choices
= m_{A} : m_{Ac}
= 3 : 3
= 1 : 1
against
Odds against getting an even integer⇒ Odds against Event A
= Number of Unfavorable Choices : Number of Favorable Choices
= m_{Ac} : m_{A}
= 3 : 3
= 1 : 1
Problem 3
Solution
Total Number of Possible Choices
= 6 {ONE, TWO, THREE, FOUR, FIVE, SIX}
⇒ n = 6
Let
A : the event of getting an odd number or a multiple of 4.
For Event A
= 4 {ONE, THREE, FOUR, FIVE}
⇒ m_{A} = 4
Probability of getting an odd number or a multiple of 4
⇒ Probability of occurrence of Event A
= 

⇒ P(A)  = 
 
= 
 
= 

Odds
Number of Unfavorable Choices= Total Number of possible choices − Number of Favorable choices
⇒ m_{A}^{c}  =  n − m_{A} 
=  6 − 4  
=  2 
in favor
Odds in Favor of getting an odd number or a multiple of 4⇒ Odds in Favor of Event A
= Number of Favorable Choices : Number of Unfavorable Choices
= m_{A} : m_{Ac}
= 4 : 2
= 2 : 1
against
Odds against getting an odd number or a multiple of 4⇒ Odds against Event A
= Number of Unfavorable Choices : Number of Favorable Choices
= m_{Ac} : m_{A}
= 2 : 4
= 1 : 2
Problem 4
Solution
Total Number 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
Number 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
= 
 
⇒ 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
Number of Unfavorable Choices
=  Total Number of possible choices − Number 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
= 
 
⇒ P(A^{c})  = 
 

= 
 
= 

Problem 5
Solution
Total Number of Possible Choices
= 6 {ONE, TWO, THREE, FOUR, FIVE, SIX}
⇒ n = 6
Let
A : the event of getting a number ≥ 4.
For Event A
= 3 {FOUR, FIVE, SIX}
⇒ m_{A} = 3
Probability of getting a number ≥ 4
⇒ Probability of occurrence of Event A
= 

⇒ P(A)  = 
 
= 
 
= 

Odds
Number of Unfavorable Choices= Total Number of possible choices − Number of Favorable choices
⇒ m_{A}^{c}  =  n − m_{A} 
=  6 − 3  
=  3 
in favor
Odds in Favor of getting a number ≥ 4⇒ Odds in Favor of Event A
= Number of Favorable Choices : Number of Unfavorable Choices
= m_{A} : m_{Ac}
= 3 : 3
= 1 : 1
against
Odds against getting a number ≥ 4⇒ Odds against Event A
= Number of Unfavorable Choices : Number of Favorable Choices
= m_{Ac} : m_{A}
= 3 : 3
= 1 : 1
Problem 6
 Getting 4 when a dice is rolled
 Getting a face having a number less than 5?
 Throwing a number greater than 2.
 The number appearing on top is not an even number.
 Getting 3 and 5 simultaneously.
 Getting 4 or 6 in a throw of single die
 An an odd number less than 4 turns up
 An ace turns up
 Getting 7
 Getting an Even number or a multiple of 3
Solution
Total Number of Possible Choices
= 6 {ONE, TWO, THREE, FOUR, FIVE, SIX}
⇒ n = 6
Let
A : the event of getting 4 when the dice is rolled
For Event A
Number of Favorable Choices= 1 {FOUR}
⇒ m_{A} = 1
Let
B : the event of getting a face having a number less than 5
For Event B
Number of Favorable Choices= 4 {ONE, TWO, THREE, FOUR}
⇒ m_{B} = 4
Let
C : the event of throwing a number greater than 2
For Event C
Number of Favorable Choices= 4 {THREE, FOUR, FIVE, SIX}
⇒ m_{C} = 4
Let
D : the event of the number appearing on top not being an even number
For Event D
Number of Favorable Choices= 3 {ONE, THREE, FIVE}
⇒ m_{D} = 3
Let
E : the event of getting 3 and 5 simultaneously.
For Event E
Number of Favorable Choices= 0 {Φ}
⇒ m_{E} = 0
Let
F : the event of getting 4 or 6 on a single throw.
For Event F
Number of Favorable Choices= 2 {FOUR, SIX}
⇒ m_{F} = 2
Let
G : the event that an odd number less than 4 turns up.
For Event G
Number of Favorable Choices= 2 {ONE, THREE}
⇒ m_{G} = 2
Let
H : the event that an ace turns up.
For Event H
Number of Favorable Choices= 1 {ONE}
⇒ m_{H} = 1
Let
I : the event of getting 7.
For Event I
Number of Favorable Choices= 0 {Φ}
⇒ m_{I} = 0
Since the die has only numbers from One to Six marked on it, the number 7 will not appear.
Let
J : the event of getting an Even number or a multiple of 3
For Event J
Number of Favorable Choices= 4 {TWO, THREE, FOUR, SIX}
⇒ m_{J} = 4
Practice Problems
 What is the chance of throwing a 5 with an ordinary dice?
 The probability of getting an odd number when we throw a single die is
 The probability of getting a number less than four when a die is rolled is __
 Find the probability of throwing a number greater than 4 when a die is rolled
 In a throw of a single die the probability of getting 3 or 5 is ___?
A dice is rolled, find
 P(even number)
 P(a number > 1)
 P(a number < 5)
 P(a number more than 6)
 P(a number < 7)
When a perfect die is rolled what is the probability of getting a face having
 4 Points
 Odd Number
 2 Points Or 3 Points
 Find the probability of getting 2 when a die is rolled
 What is the probability of throwing a number greater than 3 with an ordinary dice?
 A die is rolled. What is the probability that a number 1 or 6 may appear on the upper face?
 A (sixfaced) die is thrown. Find the chance that any one of 1, 2, 3 turns up?
If a die is tossed, what is the probability that the number appearing on top is
 even
 less than 4
 not an even number
 either an even or an odd number
 an odd number less than 4.
 The probability of not getting 1, when a die is rolled
 If a die is tossed, what is the chance of getting an even number greater than 2
 Find the chance of not throwing an ace, two or three in a single throw with a die.
 what are the odds against throwing ace or six in a single throw with a die? And what are the odds in favour?