Problem Solving : Choosing One/Single Coin Dice/Die, Card, Number, Ticket, Ball, Member, Student

Throwing/Tossing/Flipping a single (one) Coin

A coin has two faces, HEADS and TAILS.

When a coin is tossed, one of these two faces appears.

There is an equally likely chance for these two faces to appear.

Therefore, in the experiment of tossing a coin,

  • There are two possible outcomes/choices which are equally likely, mutually exclusive and exhaustive.
  • The total number of possible choices is 2 ⇒ n = 2

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Rolling/Tossing/Throwing a Single (one) Die/Dice

A dice or die unless otherwise specified has six faces, each engraved or marked with either 1, 2, 3, 4, 5 or 6 dots. Each dot is considered a number and therefore, we assume that the faces of the die or dice are marked with the numbers 1, 2, 3, 4, 5 or 6 respectively.

When a dice or die is tossed/rolled/thrown, one of these faces appears which implies that one of the six numbers appear on the face of the dice/die. Only one of the faces will appear on a single throw. All the six faces have an equally likely chance of appearance.

In the experiment of tossing/rolling/throwing a dice/die,

  • There are six possible outcomes or choices, which are equally likely, mutually exclusive and exhaustive.
  • The total number of possible choices is 6 ⇒ n = 6.

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Drawing/Picking/Selecting/Choosing a Single (one) card from a Pack of Cards

A pack of cards unless otherwise specified has 52 cards divided into four sets of 13 each.

Each set has a unique identification in the form of a symbol marked on the cards.

The four symbols used to identify/mark the cards are "Spades" (♠), "Clubs" (♣), "Diamonds" (♦) and "Hearts" (♥)

Two of the sets are colored red and two others black i.e. there are 26 (13 × 2) red cards and 26 (13 × 2) black cards.

Each set has 13 cards which are marked "A", "2", "4", "5", "6", "7", "8", "9", "10", "K", "Q", "J".

  • "A" represents the number "ONE" and is also called "Ace".
  • "K" represents kings and is marked with a picture indicative of a king.
  • "Q" represents Queen and is marked with a picture indicative of a Queen.
  • "J" represents a young prince and is marked with a picture indicative of a Young Prince. It is also called a "Jack" or a "Knave".
Spades (♠)Clubs (♣)Diamonds (♦)Hearts (♥)Total Cards
A
2
3
4
5
6
7
8
9
10
K
Q
J
A
2
3
4
5
6
7
8
9
10
K
Q
J
A
2
3
4
5
6
7
8
9
10
K
Q
J
A
2
3
4
5
6
7
8
9
10
K
Q
J
4
4
4
4
4
4
4
4
4
4
4
4
4
1313131352

Shuffle

Meaning

  1. Mix so as to make a random order or arrangement.

Well shuffled cards

Well shuffled cards implies cards that are mixed up well.

Experiment : Drawing a single/one card

In the experiment of drawing a card from a pack of 52 cards,
  • Finding total number of possible choices

    The total no. of possible choices = No. of ways in which one card can be drawn from the total 52 cards
    ⇒ n = 52C1
    Number of combinations of "52" different things taking "1" at a time.
    =
    52
    1
    = 52
    Note
    1. By logical reasoning we would be able to say that the card drawn may be any of the 52 cards. These form the total number of possible choices for the experiment of drawing a card from the pack of 52 cards.
    2. The experiment that we identify for the purpose of calculating probability would vary depending on the number of cards being drawn at a time. Drawing one card is an experiment different from the experiment of drawing two cards.
  • Finding Favourable/Favorable Choices

    To find the number of favorable choices, divide the total 52 cards into two groups.
    1. One consisting of the cards which are favourable/favorable for the occurance of the event
    2. The other consisting of all the other cards
Examples
  1. Find the probability of drawing a card which is red.

    Let,

    1. "A" : The Event of a drawing a card which is red.

    • For Event 'A'

    FavourableUnfavourable
    Red CardsOthersTotal
    Available262652
    To Choose101
    Choices26C126C052C1

    We can find the total number of possible choices for the experiment as well as the number of favourable/favorable choices for the event from the table.

    No. of favorable/favourable choices = No. of ways in which one red card can be drawn from the total 26 red cards.
    ⇒ mA = 26C1
    Number of combinations of "26" different things taking "1" at a time.
    =
    26
    1
    = 26

  2. Find the probability of drawing a card which is a spades or a king.

    There are

    • 4 kings, one of each type (spades, clubs, hearts and diamonds).
    • 13 spades.
    • The 13 spades include the king of spades.
      ⇒ There would be 3 kings (of clubs, hearts and diamonds) in the remaining cards.

    The set of cards from which any card can appear to make our effort a success would include the 13 spades and the 3 other kings.

    Let,

    1. "K" : The Event of a drawing a card which is a spades or a king.

    • For Event 'K'

    FavourableUnfavourable
    Spades and KingsOthersTotal
    Available16 (13 + 3)3652
    To Choose101
    Choices16C136C052C1

    We can find the total number of possible choices for the experiment as well as the number of favourable/favorable choices for the event from the table.

    No. of favorable/favourable choices = No. of ways in which one card which is either a spade or a king can be drawn from the total 16 cards
    ⇒ mK = 16C1
    Number of combinations of "16" different things taking "1" at a time.
    =
    16
    1
    = 16

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Drawing/Picking/Choosing/Selecting a Single/One entity/item/object from a group

We will come across a number of problems involving picking up one entity from a group. For example
  • Choosing a Number from a group/set of Numbers
  • Choosing a Coloured/Colored ball from a number of balls
  • Choosing a ticket from tickets marked with numbers
  • Choosing a page from a book whose pages are numbered
  • Choosing a person from among a group of persons

Problem Solving Logic

A logic similar to the one adopted for solving problems where a card is drawn from a pack of cards is adopted.

Experiment : Drawing a single/one entity/item

In the experiment of drawing/choosing/picking/selecting an entity/item from a set of entities/items,
  • Finding total number of possible choices
    The total no. of possible choices = No. of ways in which one item can be drawn from the total 'x' items
    ⇒ n = xC1
    Number of combinations of "x" different things taking 1 at a time.
    =
    n
    1
    = n
  • Finding Favourable/Favorable Choices
    To find the number of favorable choices, divide the total x entities/items into two groups.
    1. One consisting of the entities/items which are favourable/favorable for the occurance of the event.
    2. The other consisting of all the other entities/items.

Build a working table with that data.

FavorableUnfavorable
Item DescriptionOthersTotal
Availableabx (a + b)
To Choose101
ChoicesaC1bC0xC1

We can find the total number of possible choices for the experiment as well as the number of favourable/favorable choices for the event from the table.

Examples

  1. Find the probability of drawing a red or a black ball from a bag contains 3 red balls, 4 white balls and 7 black balls.

    Let,

    1. "A" : The Event of a drawing a red or black ball.

    • For Event 'A'

    FavourableUnfavourable
    [Red + Black]OthersTotal
    Available10414
    To Choose101
    Choices10C14C014C1

    We can find the total number of possible choices for the experiment as well as the number of favourable/favorable choices for the event from the table.

    No. of favorable/favourable choices = No. of ways in which one ball which is either red or black can be drawn from the total 10.
    ⇒ mA = 10C1
    Number of combinations of "10" different things taking "1" at a time.
    =
    10
    1
    = 10

... 121314 ...
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