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

Page PS2

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

= 2 {FOUR, SIX}
⇒ mA = 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

= Total Number of possible choices − Number of Favorable choices
⇒ mAc = n − mA
= 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?
Author : The Edifier
Page PS4