Identifying the experiment, defining the event and finding the number of Favorable Choices in Drawing/Picking/Choosing/Selecting a Numbered Card from a Pack/Box/Urn

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Problem 9

Define the Event and identify the number of favourable choices in the following cases:
  1. Drawing a card with a number which is a square marked on it from a packet of 100 cards numbered 1 to 100.
  2. Choosing a number at random from among the first 120 natural numbers which is a multiple of 5 or 15
  3. Drawing a card with a number that is divisible by 3 or 7 on drawing a card from a set of 17 cards numbered 1, 2, 3, ... , 17.
  4. Choosing at random a number that is divisible by 6 or 8 from among 1 to 90
  5. Getting a number between 1 and 100 which is divisible by 1 and itself only is ...
Ans : 1) 10, 2) 24, 3) 5, 4) 23, 5) 13

Solution

  1. Experiment :

    Drawing a card from a packet of 100 cards numbered 1 to 100

    Total number of cards

    = 100

    Let

    A : the event of drawing a card with a number which is a square marked on it

    For Event A

    Number of Numbers which are squares

    = 10

    {1, 4, 9, 16, 25, 36, 49, 64, 81, 100}

    Favorable
    (Squares)     		Unfavorable
    (Others)     		Total
    Available 10 90 100
    To Choose 1 0 1
    Choices 10C190C0100C1

    Number of Favorable Choices

    = Number of ways in which a card with a number which is a square can be drawn from the total 10 favorable cards

    ⇒ mA = 10C1
    =
    10
    1
    = 10

  2. Experiment :

    Choosing a number from among the first 120 natural numbers

    Total number of numbers

    = 120

    Let

    B : the event of choosing a number which is a multiple of 5 or 15

    For Event B

    From among the first 120 natural numbers

    Multiples of 5 ⇒ 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100, 105, 110, 115, 120

    Multiples of 15 ⇒ 15, 30, 45, 60, 75, 90

    The numbers which are LCM of the two numbers and the multiples of LCM repeat. The remaining numbers form the required numbers.

    15 is the LCM of 5 and 15 and multiples of 15 appear in both the multiple's list.

    Number of Numbers which are Multiples of 5 or 15 (= multiples of 5)

    = 24

    {5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100, 105, 110, 115, 120}

    Favorable
    (Multiples of 5 or 15)     		Unfavorable
    (Others)     		Total
    Available 24 96 120
    To Choose 1 0 1
    Choices 24C196C0120C1

    Number of Favorable Choices

    = Number of ways in which a number which is a multiple of 5 or 15 can be chosen from the total 24 favorable numbers

    ⇒ mB = 24C1
    =
    24
    1
    = 24

  3. Experiment :

    Drawing a card from a set of cards numbered from 1 to 13

    Total number of cards

    = 13

    Let

    C : the event of drawing a card with a number that is a divisible by 3 or 5

    The numbers divisible by a are the multiples of a.

    All multiples of 3 are divisible by 3 and all multiples of 7 are divisible by 7.

    For Event C

    From 1, 2, ..., 13

    Multiples of 3 ⇒ 3, 6, 9, 12

    Multiples of 7 ⇒ 7

    Multiples of 3 or 7 ⇒ 3, 6, 7, 9, 12

    The numbers which are LCM of the two numbers and the multiples of LCM repeat. The remaining numbers form the required numbers.

    There are no common multiples for 3 and 7 from 1 to 13

    Number of Numbers which are divisible by 3 or 7 (= multiples of 3 or 7)

    = 5

    {3, 6, 7, 9, 12}

    Favorable
    (divisible by 3 or 7)     		Unfavorable
    (Others)     		Total
    Available 5 8 13
    To Choose 1 0 1
    Choices 5C18C013C1

    Number of Favorable Choices

    = Number of ways in which a number which is divisble by 3 or 7 can can be drawn from the total 5 favorable numbers

    ⇒ mC = 5C1
    =
    5
    1
    = 5

  4. Experiment :

    Choosing a random a number from among 1 to 90

    Total number of numbers

    = 90

    Let

    D : the event of choosing a random number that is divisible by 6 or 8

    The numbers divisible by a are the multiples of a.

    All multiples of 6 are divisible by 6 and all multiples of 8 are divisible by 8.

    For Event D

    From 1, 2, ..., 90

    Multiples of 6 ⇒ 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90

    Multiples of 8 ⇒ 8, 16, 24, 32, 40 48, 56, 64, 72, 80, 88

    Multiples of 6 or 8 ⇒ 6, 8, 12, 16, 18, 24, 30, 32, 36, 40, 42, 48, 54, 56, 60, 64, 66, 72, 78, 80, 84, 88, 90

    The numbers which are LCM of the two numbers and the multiples of LCM repeat. The remaining numbers form the required numbers.

    24 which is the LCM of 6 and 8 and multiples of 24 appear in both the multiple's list.

    Number of Numbers which are divisible by 6 or 8 (= multiples of 6 or 8)

    = 235

    {6, 8, 12, 16, 18, 24, 30, 32, 36, 40, 42, 48, 54, 56, 60, 64, 66, 72, 78, 80, 84, 88, 90}

    Favorable
    (divisible by 6 or 8)     		Unfavorable
    (Others)     		Total
    Available 23 67 90
    To Choose 1 0 1
    Choices 23C167C090C1

    Number of Favorable Choices

    = Number of ways in which a number which is divisble by 6 or 8 can can be drawn from the total 23 favorable numbers

    ⇒ mD = 23D1
    =
    23
    1
    = 23

  5. Experiment :

    Picking a number between 5 and 55

    Total number of numbers

    = 51

    Let

    E : the event of picking a number that is divisible by 1 and itself only

    Numbers divisible by 1 and itself are prime numbers

    For Event E

    Number of Numbers which are divisible by 1 and itself

    = 13

    {7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53}

    Favorable
    (divisible by 1 and itself)     		Unfavorable
    (Others)     		Total
    Available 13 38 51
    To Choose 1 0 1
    Choices 13C138C051C1

    Number of Favorable Choices

    = Number of ways in which a number which is divisible by 1 and itself only can be picked from the total 13 favorable cards

    ⇒ mE = 13C1
    =
    13
    1
    = 13
Author : The Edifier
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