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

## 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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- Problems Solutions
- 3_1. Total number of possible choices/events/outcomes in tossing/throwing/rolling a die/dice
- 3_2. Chance for an even number more than 2 turning up on throwing a six faced die/dice
- 3_3. Probability and Odds for getting an even integer
- 3_4. Probability and Odds in favor/favour of getting an odd number or a multiple of 4 on tossing/rolling a die/dice
- 3_5. Probability and odds for getting a number greater than equal to 4 on throwing a die/dice once
- 3_6. Define the event and identify the number of favorable/favourable choices in the experiment of tossing a die/dice

## 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.

### Shuffle

#### Meaning

#### 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, **Examples**

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 |

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

#### 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 = ^{52}C_{1}Number of combinations of "52" different things taking "1" at a time.= 52 1 = 52 ##### Note

- 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.
- 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.- One consisting of the cards which are favourable/favorable for the occurance of the event
- The other consisting of all the other cards

- Find the probability of drawing a card which is red.
Let,

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

### • For Event 'A'

Favourable Unfavourable Red Cards Others Total Available 26 26 52 To Choose 1 0 1 Choices ^{26}C_{1}^{26}C_{0}^{52}C_{1}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. ⇒ m _{A}= ^{26}C_{1}Number of combinations of "26" different things taking "1" at a time.= 26 1 = 26 - 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,

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

### • For Event 'K'

Favourable Unfavourable Spades and Kings Others Total Available 16 (13 + 3) 36 52 To Choose 1 0 1 Choices ^{16}C_{1}^{36}C_{0}^{52}C_{1}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 ⇒ m _{K}= ^{16}C_{1}Number of combinations of "16" different things taking "1" at a time.= 16 1 = 16

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- Problems Solutions
- 4_1. Probability and Odds for a card drawn from a pack of cards being an ace of club
- 4_2. Define the event and identify the number of favorable/favourable choices in drawing a card from a pack of cards
- 4_3. Total number of possible choices/events/outcomes in picking/choosing/selecting/drawing a card from a pack of cards
- 4_4. Probability and Odds for a card drawn from a pack of cards being either a spade or a king
- 4_5. Probability and Odds of not getting a jack or a king on drawing a card from a pack of cards
- 4_6. Odds against a gambler winning a bet that a card drawn from a pack of cards is a spade or an ace

## 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 ### Problem Solving Logic

### Examples

- 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

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 = ^{x}C_{1}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.- One consisting of the entities/items which are favourable/favorable for the occurance of the event.
- The other consisting of all the other entities/items.

Build a working table with that data.

Favorable | Unfavorable | ||
---|---|---|---|

Item Description | Others | Total | |

Available | a | b | x (a + b) |

To Choose | 1 | 0 | 1 |

Choices | ^{a}C_{1} | ^{b}C_{0} | ^{x}C_{1} |

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.

- 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,

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

### • For Event 'A'

Favourable Unfavourable [Red + Black] Others Total Available 10 4 14 To Choose 1 0 1 Choices ^{10}C_{1}^{4}C_{0}^{14}C_{1}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. ⇒ m _{A}= ^{10}C_{1}Number of combinations of "10" different things taking "1" at a time.= 10 1 = 10

... 121314 ... |

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- Problems Solutions
- 5_1. Number of successes and failures for a ball drawn from a bag containing 5 white and 3 black balls being black
- 5_2. probability and odds that a ball drawn from a bag containing different colored balls is of a particular colour
- 5_3. probability of drawing a ball of one of the two colors from a bag containing three different colored balls
- 5_4. probability and odds that a ball drawn from a bag containing three different colored balls is not of one or more specified colors
- 5_5. Define the event and identify the number of favorable/favourable choices in drawing a ball from balls of two or more colors
- 6_1. Picking a numbered ball with a number which is a multiple of either of the two numbers
- 6_2. Define the event and identify the number of favorable/favourable choices in drawing a numbered card from a bag containing cards with numbers on them
- 6_3. Choosing a digit from the given digits
- 6_4. Choosing a single card with a prime number less than 20
- 6_5. Drawing a numbered ticket and getting specified sum of digits product of digits
- 6_6. Selecting a numbered card with a number which is a multiple of two or more numbers
- 6_7. Integer chosen from a set of integers being divisible by two given integers
- 6_8. Selecting a numbered card with a number having a specified sum of the digits
- 6_9. A number picked at random from a set of numbers satisfying a certain rule
- 7_1. Opening a page from the pages of a book such that the page number has specified characteristics
- 7_2. Selecting a page from a book such that the digits of the page number have a specified sum
- 7_3. Choosing a bolt which is not defective from a box containing some defective bolds
- 7_4. Picking a product which is non defective from a lot consisting of prodcuts with major and minor defects
- 8_1. Selecting a letter that is a consonant from the letters of a word where all letters are different
- 8_2. Selecting a letter that is a vowel from the letters of a word where there are repeating letters
- 9_1. Selecting a specified letter from the letters within a book

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- Problems Solutions
- 2. Tossing/Throwing/Flipping a single Coin
- 3. Throwing/Tossing/Rolling Single/One Dice/Die
- 4. Drawing/Picking/Choosing/Selecting Single/One card from a well shuffled pack/deck of cards
- 5. Drawing/Picking/Choosing/Selecting Single/One ball from a bag/urn/box containing two or more balls
- 6. Drawing a numbered card from a pack containing two or more numbered cards
- 7. Selecting an Item/Product/Article/Page from two or more ...
- 8. Drawing/Picking/Choosing a Single/One Letter/Character from a word
- 9. Drawing/Picking/Choosing a Single/One Letter/Character from a book

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