فضاء العينة: التمثيل والتعريف

Representing Sample Spaces

• In football matches, the referee tosses a coin to decide which team chooses the field side first.
• The outcome is either the emblem (Heads - L) or the writing (Tails - T).

Defining Sample Space

• Definition: Listing, tabulating, or using tree diagrams to represent the sample space.

• Random Experiment: An action where all possible outcomes are known beforehand.

• Outcomes: All possible results of an experiment.

• Event: A single outcome or a combination of outcomes.

• Sample Space: The set of all possible outcomes.

Representing Sample Spaces

• Example: Tossing a coin twice.
• Possible outcomes for each toss: Emblem (L) or Writing (T).

List Method

• Pair each possible outcome of the first toss with all possible outcomes of the second toss.
• The sample space is denoted as: L,L , L,T , T,L , T,T.

Table Method

• List the possible outcomes of the first toss in the right column and second toss in the top row.
• Possible outcomes: L,L, L,T , T,L, T,T.

Tree Diagram Method

• First Toss (L or T), Second Toss (L or T). Results are L,L , L,T , T,L , T,T.

Multi-Stage Experiments

• The experiment conducted in Example 1 is a two-stage experiment.
• A multi-stage experiment involves conducting an experiment over several stages.

Tree Diagram for Multi-Stage Experiments

• Example: Sandwich Choices (Size, Cheese, Tomato, Pickles).

Stages

• Size: S (Small), M (Medium), L (Large).
• Cheese: C (With Cheese), NC (Without Cheese).
• Tomato: T (With Tomato), NT (Without Tomato).
• Pickles: P (With Pickles), NP (Without Pickles).

Fundamental Counting Principle

• Used to find the total number of possible outcomes.

Principle

• If an event occurs over n stages the total number of possible outcomes is found by multiplying the number of outcomes in each stage.

Example

• Saad wants to buy a thobe from the options shown in the adjacent table. How many options does he have to choose a suitable thobe?

Calculating Options

• Available (from the table): Fabrics, Colors, Sleeves, Neck Opening, Front Opening, Buttons
• Use the fundamental counting principle: 5 🞩 6 🞩 3 🞩 3 🞩 2 🞩 2 = 1080

Permutations and Combinations

• Used to calculate probabilities by considering the order of elements.

Permutation

• An arrangement of a set of elements in a specific order.

Fundamental Counting Principle

• Used to calculate the number of possible arrangements. 4 🞩 3 🞩 2 🞩 1 = 24

Factorial Notation

• n! represents the product of all positive integers less than or equal to n.

Probability with Permutations

• Example: Probability of specific arrangement in sports team numbering.
• The probability is calculated as: P (Nawaf gets 1 and Majed gets 2) = 18!/ 20!

General Case

• Consider a set of n distinct elements:
• The number of permutations of r items chosen from n is: P(n,r) = n! / (n-r)!

Compound Event

• An event containing more than one step.
• We calculate compound event probabilities using conditional probability.

Geometric Probability

• Geometric Probability: An event probability depends on the length or surface area.

Line Segment

• If you want to find the chance that a point on segment AD is also on segment BC, divide the length of BC by the length of AD.

Arcs

• You can use the ratio of its degree measure to 360 to find the arc’s area.

Overlapping Events

• Probability is calculated knowing the chance of single steps.
• This uses both events and must factor out what we know.