# Problem 1

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 SIX

⇒ The total number of possible choices in the experiment = 6

# Problem 2

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,

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

Number of Favorable Choices

= 3 {TWO, FOUR, SIX}

⇒ mA = 3

Probability of getting an an even integer on rolling a dice

⇒ Probability of occurrence of Event A

=
 Number of Favorable Choices for the Event Total Number of Possible Choices for the Experiment
⇒ P(A) =
 mA n
=
 3 6
=
 1 2

## Odds

Number of Unfavorable Choices

= Total Number of possible choices − Number of Favorable choices

 ⇒ mAc = n − mA = 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

= mA : mAc

= 3 : 3

= 1 : 1

## against

Odds against getting an even integer

⇒ Odds against Event A

= Number of Unfavorable Choices : Number of Favorable Choices

= mAc : mA

= 3 : 3

= 1 : 1

# Problem 3

Find the probability and odds in favor of getting an odd number or a multiple of 4 on throwing a dice

# Solution

In the experiment of throwing a dice,

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

Number of Favorable Choices

= 4 {ONE, THREE, FOUR, FIVE}

⇒ mA = 4

Probability of getting an odd number or a multiple of 4

⇒ Probability of occurrence of Event A

=
 Number of Favorable Choices for the Event Total Number of Possible Choices for the Experiment
⇒ P(A) =
 mA n
=
 4 6
=
 2 3

## Odds

Number of Unfavorable Choices

= Total Number of possible choices − Number of Favorable choices

 ⇒ mAc = n − mA = 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

= mA : mAc

= 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

= mAc : mA

= 2 : 4

= 1 : 2

# Problem 4

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 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

 ⇒ mA = 2 {FOUR, SIX} = 2

Probability that an even number more than 2 turns up

⇒ Probability of occurrence of Event A

=
 Number of 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

Number of Unfavorable Choices

 ⇒ mAc n − mA = Total Number of possible choices − Number 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 UnFavorable/Unfavorable Choices for the Event Total Number of Possible Choices for the Experiment
⇒ P(Ac) =
 mAc n
=
 4 6
=
 2 3

# Problem 5

A die is thrown once, find P (a number ≥ 4) and also the odds.

# Solution

In the experiment of throwing a dice,

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

Number of Favorable Choices

= 3 {FOUR, FIVE, SIX}

⇒ mA = 3

Probability of getting a number ≥ 4

⇒ Probability of occurrence of Event A

=
 Number of Favorable Choices for the Event Total Number of Possible Choices for the Experiment
⇒ P(A) =
 mA n
=
 3 6
=
 1 2

## Odds

Number of Unfavorable Choices

= Total Number of possible choices − Number of Favorable choices

 ⇒ mAc = n − mA = 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

= mA : mAc

= 3 : 3

= 1 : 1

## against

Odds against getting a number ≥ 4

⇒ Odds against Event A

= Number of Unfavorable Choices : Number of Favorable Choices

= mAc : mA

= 3 : 3

= 1 : 1

# Problem 6

Define the Event and identify the number of favourable and choices in the following which relate to the experiment of rolling a dice/die:
1. Getting 4 when a dice is rolled
2. Getting a face having a number less than 5?
3. Throwing a number greater than 2.
4. The number appearing on top is not an even number.
5. Getting 3 and 5 simultaneously.
6. Getting 4 or 6 in a throw of single die
7. An an odd number less than 4 turns up
8. An ace turns up
9. Getting 7
10. Getting an Even number or a multiple of 3

# Solution

In the experiment of throwing a dice,

Total Number of Possible Choices

= 6 {ONE, TWO, THREE, FOUR, FIVE, SIX}

⇒ n = 6

1. Let

A : the event of getting 4 when the dice is rolled

## For Event A

Number of Favorable Choices

= 1 {FOUR}

⇒ mA = 1

2. 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}

⇒ mB = 4

3. Let

C : the event of throwing a number greater than 2

## For Event C

Number of Favorable Choices

= 4 {THREE, FOUR, FIVE, SIX}

⇒ mC = 4

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}

⇒ mD = 3

5. Let

E : the event of getting 3 and 5 simultaneously.

## For Event E

Number of Favorable Choices

= 0 {Φ}

⇒ mE = 0

6. Let

F : the event of getting 4 or 6 on a single throw.

## For Event F

Number of Favorable Choices

= 2 {FOUR, SIX}

⇒ mF = 2

7. Let

G : the event that an odd number less than 4 turns up.

## For Event G

Number of Favorable Choices

= 2 {ONE, THREE}

⇒ mG = 2

8. Let

H : the event that an ace turns up.

## For Event H

Number of Favorable Choices

= 1 {ONE}

⇒ mH = 1

9. Let

I : the event of getting 7.

## For Event I

Number of Favorable Choices

= 0 {Φ}

⇒ mI = 0

Since the die has only numbers from One to Six marked on it, the number 7 will not appear.

10. 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}

⇒ mJ = 4

# Practice Problems

1. What is the chance of throwing a 5 with an ordinary dice?
2. The probability of getting an odd number when we throw a single die is
3. The probability of getting a number less than four when a die is rolled is __
4. Find the probability of throwing a number greater than 4 when a die is rolled
5. In a throw of a single die the probability of getting 3 or 5 is ___?
6. 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)
7. 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
8. Find the probability of getting 2 when a die is rolled
9. What is the probability of throwing a number greater than 3 with an ordinary dice?
10. A die is rolled. What is the probability that a number 1 or 6 may appear on the upper face?
11. A (six-faced) die is thrown. Find the chance that any one of 1, 2, 3 turns up?
12. If a die is tossed, what is the probability that the number appearing on top is

1. even
2. less than 4
3. not an even number
4. either an even or an odd number
5. an odd number less than 4.
13. The probability of not getting 1, when a die is rolled
14. If a die is tossed, what is the chance of getting an even number greater than 2
15. Find the chance of not throwing an ace, two or three in a single throw with a die.
16. what are the odds against throwing ace or six in a single throw with a die? And what are the odds in favour?