# Throwing/Tossing/Rolling Single/One Dice/Die : Probability : Problems & Solutions

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.

 Solution
 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.

 Solution
In the experiment of rolling a cubical dice/die,

 ⇒ n Total No. of Possible Choices = 6 {ONE, TWO, THREE, FOUR, FIVE, SIX} = 6

Let "A" be the event of getting an even integer.

# For Event "A"

 ⇒ mA No. of Favorable Choices = 3 {TWO, FOUR, SIX} = 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) =
 mA n
=
 3 6
=
 1 2

# • Odds

 ⇒ mAc n − mA No. of UnFavorable Choices = Total No. of possible choices − No. of Favorable choices = = 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 : mAc = 3 : 3 = 1 : 1

## » against

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

 Solution

In the experiment of throwing a dice/die,

 ⇒ n Total No. of Possible Choices = 6 {ONE, TWO, THREE, FOUR, FIVE, SIX} = 6

Let "A" be the event of getting an odd number or a multiple of 4.

# For Event "A"

 ⇒ mA No. of Favorable Choices = 4 {ONE, THREE, FOUR, FIVE} = 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) =
 mA n
=
 4 6
=
 2 3

# • Odds

 ⇒ mAc n − mA No. of UnFavorable Choices = Total No. of possible choices − No. of Favorable choices = = 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 : mAc = 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?

 Solution
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"

 ⇒ mA No. of Favorable Choices = 2 {FOUR, SIX} = 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) =
 mA n
=
 2 6
=
 1 3

Probability that an even number more than 2 does not turn up

⇒ Probability of non-occurrence of Event "A" = 1 − Probability of occurrence of Event A
P(Ac) = 1 − P(A)
=
1 −
 1 3
=
 3 − 1 3
=
 2 3

# • Alternative

 ⇒ mAc n − mA No. of UnFavorable Choices = Total No. of possible choices − No. of Favorable choices = = 6 − 2 = 4

Probability that an even number more than 2 does not turn up
⇒ Probability of non-occurrence of Event "A" =
 Number of UnFavourable/UnFavorable Choices for the Event Total Number of Possible Choices for the Experiment
⇒ P(Ac) =
 mAc n
=
 4 6
=
 2 3

No. Problems & Solutions
0.
A die is thrown once, find P (a number ≥ 4) and also the odds.

 Solution

In the experiment of throwing a dice/die once,

 ⇒ n Total No. of Possible Choices = 6 {ONE, TWO, THREE, FOUR, FIVE, SIX} = 6

Let "A" be the event of getting a number ≥ 4.

# For Event "A"

 ⇒ mA No. of Favorable Choices = 3 {FOUR, FIVE, SIX} = 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) =
 mA n
=
 3 6
=
 1 2

# • Odds

 ⇒ mAc n − mA No. of UnFavorable Choices = Total No. of possible choices − No. of Favorable choices = = 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 : mAc = 3 : 3 = 1 : 1

## » against

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

 Solution

In the experiment of tossing a dice/die

 ⇒ n Total No. of Possible Choices = 6 {ONE, TWO, THREE, FOUR, FIVE, SIX} = 6
a) Let "A" be the event of getting 4 when a die is rolled

# For Event "A"

 ⇒ mA No. of Favorable Choices = 1 {FOUR} = 1
b) Let "B" be the event of getting a face having a number less than 5 on tossing a die

# For Event "B"

 ⇒ mB No. of Favorable Choices = 4 {ONE, TWO, THREE, FOUR} = 4
c) Let "C" be the event of throwing a number greater than 2

# For Event "C"

 ⇒ mC No. of Favorable Choices = 4 {THREE, FOUR, FIVE, SIX} = 4
d) Let "D" be the event of the number appearing on the die not being an even number

# For Event "D"

 ⇒ mD No. of Favorable Choices = 3 {ONE, THREE, FIVE} = 3
e) Let "E" be the event of getting 3 and 5 simultaneously

# For Event "E"

 ⇒ mE No. of Favorable Choices = 0 {Φ} = 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"

 ⇒ mF No. of Favorable Choices = 2 {FOUR, SIX} = 2
g) Let "G" be the event of an odd number less than 4 turning up

# For Event "G"

 ⇒ mG No. of Favorable Choices = 2 {ONE, THREE} = 2
h) Let "H" be the event that an ACE turns up

# For Event "H"

 ⇒ mH No. of Favorable Choices = 1 {ONE} = 1
i) Let "I" be the event of getting 7

# For Event "I"

 ⇒ mI No. of Favorable Choices = 0 {Φ} = 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"

 ⇒ mJ No. of Favorable Choices = 4 {TWO, THREE, FOUR, SIX} = 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 (six-faced) 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?
 Author Credit : The Edifier