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

Throwing/Tossing a Single (one) Coin

A coin has two faces, HEADS and TAILS. When a coin is tossed, one of these faces appears. There is an equally likely chance for these two faces to appear.
⇒ There are two possible outcomes or choices in the experiment of tossing a coin, which are equally likely, mutually exclusive and exhaustive.
⇒ For the experiment of tossing a coin, the total number of possible choices is 2 ⇒ n = 2.

Rolling/Tossing 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.

There is an equally likely chance for all these six faces i.e. the six numbers to appear. Moreover, only one of these will appear on a single throw.
⇒ There are six possible outcomes or choices in the experiment of tossing/rolling/throwing a dice/die, which are equally likely, mutually exclusive and exhaustive.
⇒ For the experiment of tossing/throwing/rollling a dice/die, the total number of possible choices is 6 ⇒ n = 6.

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

A pack of cards unless otherwise specified has 52 cards. The 52 cards are 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
13 13 13 13 52

• Well shuffled Cards

Well shuffled cards implies cards that are mixed up well.
  • Shuffle = Mix so as to make a random order or arrangement.

• Experiment

The experiment that we identify for the purpose of calculating probability in case of drawing cards 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. As the number of cards drawn vary, the experiment varies.

• Experiment of drawing a single card

We deal with the experiment of drawing a single card here. In this experiment, i.e. In the experiment of drawing a card from a pack of 52 cards,

The total number of possible choices = Number of ways in which one card can be drawn from the total 52 cards
⇒ n = 52C1 Number of combinations of "n" different things taking "r" at a time.
=
52
1
= 52
By sheer logic you would be able to say that the card drawn may be any of the 52 cards which creates the 52 different possibilities or choices. These possibilities form the total number of possible choices for the experiment of drawing a card from the pack.

• Favourable/Favorable Choices

The number of favourable/favorable choices for drawing a required card can also be calculated by a similar logic. To enable the calculation of the number of favourable/favorable choices,
  1. Divide the total 52 cards into two groups.
  2. One group consisting of all the cards which are favourable/favorable for the occurance of the given event
  3. The second group consisting of all the other cards
Say for example, you are required to find the probability of drawing a card which is red.
Favourable/Favorable
[Red Cards]		 Others Total
Available 26 26 52
To Choose 1 0 1
Choices 26C1 26C0 52C1

Notice that you 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.

The number of favourable/favorable choices = Number of ways in which one card which is a red card can be drawn from the total 26 cards
⇒ n = 26C1 Number of combinations of "n" different things taking "r" at a time.
=
26
1
= 26
To find the probability of drawing a card which is a spades or a king
Favourable/Favorable
[Spades and Kinds]		 Others Total
Available 16 (13 + 3) 36 52
To Choose 1 0 1
Choices 16C1 36C0 52C1
There are 13 spades which includes a king in it. Therefore, we consider the other 3 kings also to give us the set of cards from which any card can appear to make our effort a success.

Notice that you 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.

The number of favourable/favorable choices = Number of ways in which one card which is either a spade or a king can be drawn from the total 16 cards
⇒ n = 16C1 Number of combinations of "n" different things taking "r" at a time.
=
16
1
= 16

• Advice

Please follow the above procedure of building up a table for finding out the number of favourable/favorable choices. It would help you a lot in solving majority of the problems based on the mathematical definition of probability.

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. An example of some of the groups that you would be encountering
  • 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.
  1. Divide the total entites into two groups.
  2. One group consisting of all the entities which are favourable/favorable for the occurance of the given event
  3. The second group consisting of all the other entities
Build a table similar to the one above.
Favourable/Favorable
[??]		 Others Total
Available a b a+b
To Choose 1 0 1
Choices aC1 bC0 (a+b)C1

This table helps you in finding the solution fast. Notice that you 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.

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