Probability – - Examples For Sample Space

Probability – Examples for Sample Space

Definition of sample space
Example: Tossing a fair coin
Example: Rolling a fair six-sided die
Example: Selecting a card from a standard deck
Example: Drawing a ball from an urn with colored marbles

Probability – Events and Their Complement

Definition of an event in probability
Complement of an event
Example: Event of getting a head when flipping a coin
Example: Event of rolling an even number with a die
Example: Event of drawing a red card from a deck
Example: Event of selecting a white marble from an urn

Probability – Union and Intersection of Events

Definition of union and intersection of events
Example: Union of two events
Example: Intersection of two events
Example: Rolling an even number or getting a head
Example: Drawing a red card and selecting a queen

Probability – Addition Rule

Addition rule for finding the probability of the union of two events
Example: Probability of getting a head or rolling an even number
Example: Probability of drawing a red card or selecting a queen
Example: Probability of drawing a spade or an ace

Probability – Conditional Probability

Definition of conditional probability
Example: Probability of getting a head given that the coin is fair
Example: Probability of rolling an even number given that the die is fair
Example: Probability of drawing a red card given that a heart is drawn

Probability – Independent Events

Definition of independent events
Example: Tossing a fair coin twice
Example: Rolling a fair die twice
Example: Drawing two cards from a deck without replacement

Probability – Multiplication Rule

Multiplication rule for finding the probability of the intersection of two independent events
Example: Probability of getting two heads when tossing a fair coin twice
Example: Probability of rolling a six twice with a fair die
Example: Probability of drawing two aces from a deck without replacement

Probability – Conditional Probability II

Revised definition of conditional probability with independent events
Example: Probability of getting a head on the first toss and a tail on the second toss
Example: Probability of rolling a four on the first roll and a five on the second roll
Example: Probability of drawing a red card on the first draw and a black card on the second draw

Probability – Combinations and Permutations

Introduction to combinations and permutations
Definition of combinations and permutations
Example: Selecting a committee of 3 members from a group of 10
Example: Arranging the letters of the word “MATHS”
Example: Choosing 2 books from a shelf of 5 books

Probability – Combinations and Permutations II

Counting principle for combinations and permutations
Example: Selecting a committee of 2 men and 1 woman from a group of 5 men and 4 women
Example: Arranging the letters of the word “MATHS” without repetition
Example: Assigning 3 tasks to 4 individuals

Probability – Examples for Sample Space

Sample space: set of all possible outcomes of a random experiment
Example: Tossing a fair coin
- Sample space: {H, T}
- H denotes head, T denotes tail
Example: Rolling a fair six-sided die
- Sample space: {1, 2, 3, 4, 5, 6}
Example: Selecting a card from a standard deck
- Sample space: {2♠, 3♠, 4♠, …, A♣, 2♣, 3♣, …, A♦, 2♦, 3♦, …, A♥, 2♥, 3♥, …, A♠}

Probability – Events and Their Complement

Event: a subset of the sample space, consisting of one or more outcomes
Complement of an event: all outcomes that are not in the event
Example: Event of getting a head when flipping a coin
- Event: {H}
- Complement: {T}
Example: Event of rolling an even number with a die
- Event: {2, 4, 6}
- Complement: {1, 3, 5}
Example: Event of drawing a red card from a deck
- Event: {2♦, 3♦, …, A♥, 2♥, 3♥, …, A♠}
- Complement: {2♠, 3♠, …, A♣, 2♣, 3♣, …, A♦}

Probability – Union and Intersection of Events

Union of events: all outcomes that belong to at least one of the events
Intersection of events: all outcomes that belong to both of the events
Example: Union of two events
- A: Event of getting a head
- B: Event of rolling an even number
- Union (A ∪ B): {H, 2, 4, 6}
Example: Intersection of two events
- A: Event of drawing a red card
- B: Event of selecting a queen
- Intersection (A ∩ B): {Q♦, Q♥, Q♠}

Probability – Addition Rule

Addition rule for finding the probability of the union of two events
- P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
Example: Probability of getting a head or rolling an even number
- P(H ∪ E) = P(H) + P(E) - P(H ∩ E) = 1/2 + 1/2 - 0 = 1
Example: Probability of drawing a red card or selecting a queen
- P(R ∪ Q) = P(R) + P(Q) - P(R ∩ Q) = 26/52 + 4/52 - 2/52 = 28/52 = 7/13

Probability – Conditional Probability

Conditional probability: the probability of an event occurring given that another event has already occurred
Example: Probability of getting a head given that the coin is fair
- P(H|F) = 1/2
- F denotes the event that the coin is fair
Example: Probability of rolling an even number given that the die is fair
- P(E|F) = 1/2
- F denotes the event that the die is fair
Example: Probability of drawing a red card given that a heart is drawn
- P(R|H) = 26/51
- H denotes the event that a heart is drawn

Probability – Independent Events

Independent events: the occurrence of one event does not affect the probability of the occurrence of another event
Example: Tossing a fair coin twice
- The outcome of the first toss does not affect the outcome of the second toss
Example: Rolling a fair die twice
- The outcome of the first roll does not affect the outcome of the second roll
Example: Drawing two cards from a deck without replacement
- The probability of drawing the second card is affected by the outcome of the first draw

Probability – Multiplication Rule

Multiplication rule for finding the probability of the intersection of two independent events
- P(A ∩ B) = P(A) * P(B)
Example: Probability of getting two heads when tossing a fair coin twice
- P(H1 ∩ H2) = P(H1) * P(H2) = 1/2 * 1/2 = 1/4
Example: Probability of rolling a six twice with a fair die
- P(6_1 ∩ 6_2) = P(6_1) * P(6_2) = 1/6 * 1/6 = 1/36

Probability – Conditional Probability II

Revised definition of conditional probability with independent events
- P(A|B) = P(A)
- If A and B are independent events, the probability of event A occurring given that event B has occurred is equal to the probability of event A occurring
Example: Probability of getting a head on the first toss and a tail on the second toss
- P(H1 ∩ T2) = P(H1) * P(T2) = 1/2 * 1/2 = 1/4
Example: Probability of rolling a four on the first roll and a five on the second roll
- P(4_1 ∩ 5_2) = P(4_1) * P(5_2) = 1/6 * 1/6 = 1/36

Probability – Combinations and Permutations

Introduction to combinations and permutations
- Combinations: when the order does not matter
- Permutations: when the order does matter
Definition of combinations and permutations
- Combinations: selecting items from a set without regard to the order
- Permutations: arranging items from a set in a specific order
Example: Selecting a committee of 3 members from a group of 10
Example: Arranging the letters of the word “MATHS”
Example: Choosing 2 books from a shelf of 5 books

Probability – Combinations and Permutations II

Counting principle for combinations and permutations
- Combinations: nCr = n! / (r!(n-r)!)
- Permutations: nPr = n! / (n-r)!
Example: Selecting a committee of 2 men and 1 woman from a group of 5 men and 4 women
- Number of ways to select 2 men from 5: 5C2 = 5! / (2!(5-2)!) = 10
- Number of ways to select 1 woman from 4: 4C1 = 4! / (1!(4-1)!) = 4
- Total number of ways: 10 * 4 = 40
Example: Arranging the letters of the word “MATHS” without repetition
- Number of ways to arrange the letters: 5P5 = 5! / (5-5)! = 120
Example: Assigning 3 tasks to 4 individuals
- Number of ways to assign the tasks: 4P3 = 4! / (4-3)! = 24

Probability – Expected Value

Definition of expected value in probability
Formula for calculating the expected value: E(X) = Σ(x * P(x))
Example: Rolling a fair six-sided die
- Probability of rolling a 1: 1/6
- Probability of rolling a 2: 1/6
- Probability of rolling a 3: 1/6
- Probability of rolling a 4: 1/6
- Probability of rolling a 5: 1/6
- Probability of rolling a 6: 1/6
- Expected value = (1 * 1/6) + (2 * 1/6) + (3 * 1/6) + (4 * 1/6) + (5 * 1/6) + (6 * 1/6) = (1/6) + (2/6) + (3/6) + (4/6) + (5/6) + (6/6) = 21/6 = 3.5
Example: Playing a game with a fair deck of cards
- Probability of drawing a red card: 1/2
- Probability of drawing a black card: 1/2
- Expected value = (1 * 1/2) + (0 * 1/2) = 1/2 + 0 = 1/2 = 0.5

Probability – Variance and Standard Deviation

Definition of variance in probability
Formula for calculating the variance: Var(X) = Σ((x - μ)^2 * P(x))
Definition of standard deviation in probability
Formula for calculating the standard deviation: σ(X) = √Var(X)
Example: Rolling a fair six-sided die
- Expected value (from previous slide): μ = 3.5
- Variance = ((1-3.5)^2 * 1/6) + ((2-3.5)^2 * 1/6) + … + ((6-3.5)^2 * 1/6) = (2.5^2 * 1/6) + (1.5^2 * 1/6) + … + (2.5^2 * 1/6) = (6.25 * 1/6) + (2.25 * 1/6) + … + (6.25 * 1/6) = 5.25
- Standard deviation = √5.25 ≈ 2.29
Example: Playing a game with a fair deck of cards
- Expected value (from previous slide): μ = 0.5
- Variance = ((1-0.5)^2 * 1/2) + ((0-0.5)^2 * 1/2) = (0.5^2 * 1/2) + (0.5^2 * 1/2) = 0.25
- Standard deviation = √0.25 = 0.5

Probability – Normal Distribution

Definition of normal distribution in probability
Characteristics of a normal distribution
- Symmetric bell-shaped curve
- Mean = median = mode
- 68-95-99.7 rule: 68% of the data falls within 1 standard deviation from the mean, 95% falls within 2 standard deviations, and 99.7% falls within 3 standard deviations
Example: Heights of adult males
- Mean = 5 feet 9 inches
- Standard deviation = 2 inches
- 68% of adult males have heights between 5 feet 7 inches and 5 feet 11 inches
- 95% of adult males have heights between 5 feet 5 inches and 6 feet 1 inch
- 99.7% of adult males have heights between 5 feet 3 inches and 6 feet 3 inches
Example: Grades on a standardized test
- Mean = 75
- Standard deviation = 10
- 68% of students scored between 65 and 85
- 95% of students scored between 55 and 95
- 99.7% of students scored between 45 and 105

Probability – Binomial Distribution

Definition of binomial distribution in probability
Characteristics of a binomial distribution
- Fixed number of independent trials
- Two possible outcomes: success or failure
- Constant probability of success on each trial
- Trials are independent of each other
Example: Flipping a fair coin 10 times
- Probability of getting heads on each flip: 0.5
- Probability of getting tails on each flip: 0.5
- P(X = k) = nCr * p^k * (1-p)^(n-k)
- P(X = 5) = 10C5 * (0.5)^5 * (0.5)^(10-5) = 252 * 0.03125 * 0.03125 ≈ 0.2461
Example: Rolling a fair six-sided die 20 times
- Probability of rolling a six on each roll: 1/6
- Probability of not rolling a six on each roll: 5/6
- P(X = k) = nCr * p^k * (1-p)^(n-k)
- P(X = 10) = 20C10 * (1/6)^10 * (5/6)^(20-10) ≈ 0.1178

Probability – Poisson Distribution

Definition of Poisson distribution in probability
Characteristics of a Poisson distribution
- Model for the number of events that occur in a fixed interval of time or space
- Events occur independently of each other
- Events occur with a known average rate
- The probability of more than one event occurring in a very short interval is negligible
Example: Number of cars that pass through a toll booth in 1 hour
- Average rate: 40 cars per hour
- P(X = k) = (λ^k * e^(-λ)) / k!
- P(X = 50) = (0.4^50 * e^(-0.4)) / 50! ≈ 0.0041
Example: Number of typographical errors in a 100-page book
- Average rate: 3 errors per page
- P(X = k) = (λ^k * e