Probability - Problem 6- Advance Ball Problem Question

Slide 1: Probability

Introduction to Probability
Definition of Probability
Concepts of Random Experiments and Events
Sample Space and Event Space
Basic Terminologies in Probability

Slide 2: Sample Space

Definition of Sample Space
Examples of Sample Space
- Coin Toss: {H, T}
- Rolling a Die: {1, 2, 3, 4, 5, 6}
Finite and Infinite Sample Spaces
Discrete and Continuous Sample Spaces
Empty Sample Space

Slide 3: Events

Definition of Events
Examples of Events
- Coin Toss: Head (H)
- Rolling a Die: Odd Number
Simple and Compound Events
Complementary Events
Mutually Exclusive and Non-Mutually Exclusive Events

Slide 4: Probability of an Event

Definition of Probability of an Event
Calculating Probability: P(E) = Number of favorable outcomes / Total number of outcomes
Probability as a Fraction, Decimal, and Percentage
Probability as a Measure of Uncertainty
Probability of Impossible and Certain Events

Slide 5: Laws of Probability

Addition Law of Probability
- P(A or B) = P(A) + P(B) - P(A and B)
- Mutually Exclusive Events
- Non-Mutually Exclusive Events
Multiplication Law of Probability
- P(A and B) = P(A) * P(B|A)
- Independent Events
- Dependent Events

Slide 6: Conditional Probability

Definition of Conditional Probability
Calculating Conditional Probability: P(A|B) = P(A and B) / P(B)
Interpreting Conditional Probability
Multiplication Rule for Independent Events
Bayes’ Theorem

Slide 7: Permutations

Introduction to Permutations
Factorial Notation (n!)
Permutations of n Objects Taken r at a Time: P(n, r)
Permutations of Distinct Objects
Permutations with Repetition

Slide 8: Permutations - Examples

Example: Arranging Letters in a Word
Example: Permutations with Repetition
Example: Permutations of a Subset

Slide 9: Combinations

Introduction to Combinations
Combinations of n Objects Taken r at a Time: C(n, r)
Combinations of Distinct Objects
Combinations with Repetition
Properties of Combinations

Slide 10: Combinations - Examples

Example: Selecting a Committee
Example: Combinations with Repetition
Example: Combinations using Pascal’s Triangle

Slide 11: Probability - Problem 1

There are 3 red balls and 2 blue balls in a bag. What is the probability of randomly selecting a red ball?
Sample Space: {R1, R2, R3, B1, B2}
Number of favorable outcomes: 3 (R1, R2, R3)
Total number of outcomes: 5 (R1, R2, R3, B1, B2)
Probability of selecting a red ball: P(R) = 3/5

Slide 12: Probability - Problem 2

A fair die is rolled. What is the probability of getting an odd number or a multiple of 3?
Sample Space: {1, 2, 3, 4, 5, 6}
Number of favorable outcomes for odd number: 3 (1, 3, 5)
Number of favorable outcomes for multiple of 3: 2 (3, 6)
Number of favorable outcomes for odd number or multiple of 3: 4 (1, 3, 5, 6)
Total number of outcomes: 6
Probability of getting an odd number or a multiple of 3: P(Odd or Multiple of 3) = 4/6 = 2/3

Slide 13: Probability - Problem 3

A bag contains 4 red balls and 5 blue balls. Two balls are drawn at random without replacement. Find the probability that both balls are red.
Sample Space: {R1, R2, R3, R4, B1, B2, B3, B4, B5}
Number of favorable outcomes: 4C2 = 6 (R1R2, R1R3, R1R4, R2R3, R2R4, R3R4)
Total number of outcomes: 9C2 = 36
Probability of both balls being red: P(2 Red Balls) = 6/36 = 1/6

Slide 14: Probability - Problem 4

Bag A contains 3 red balls and 2 blue balls, while Bag B contains 2 red balls and 4 blue balls. A fair coin is flipped. If it lands heads, a ball is drawn from Bag A; if it lands tails, a ball is drawn from Bag B. What is the probability of drawing a red ball?
Sample Space: {A, B}
Number of favorable outcomes for red ball: 3/5 (Bag A) + 2/6 (Bag B)
Probability of drawing a red ball: P(Red) = 1/2 * (3/5) + 1/2 * (2/6) = 13/30

Slide 15: Probability - Problem 5

A bag contains 5 white balls and 7 black balls. Two balls are drawn at random with replacement. Find the probability that both balls are white.
Sample Space: {W1, W2, W3, W4, W5, B1, B2, B3, B4, B5, B6, B7}
Number of favorable outcomes: 5C2 = 10 (W1W2, W1W3, W1W4, W1W5, W2W3, W2W4, W2W5, W3W4, W3W5, W4W5)
Total number of outcomes: 12C2 = 66
Probability of both balls being white: P(2 White Balls) = 10/66 = 5/33

Slide 16: Probability - Problem 6

Advance ball problem Question
- Example Question: There are 20 different balls numbered 1 to 20. A child randomly selects 3 balls without replacement. What is the probability of selecting a ball with a prime number?
- Number of favorable outcomes: 8 (2, 3, 5, 7, 11, 13, 17, 19)
- Total number of outcomes: 20C3
- Probability of selecting a ball with a prime number: P(Prime number) = 8/(20C3)

Slide 17: Trigonometry

Introduction to Trigonometry
Basics of Trigonometry
Right-Angled Triangle and Trigonometric Ratios
Sine, Cosine, and Tangent Ratios
Reciprocal and Pythagorean Identities

Slide 18: Trigonometric Ratios

Definition of Trigonometric Ratios
Sine (sin), Cosine (cos), and Tangent (tan)
Cosecant (cosec), Secant (sec), and Cotangent (cot)
Evaluating Trigonometric Ratios using Calculator
Unit Circle and Trigonometric Ratios

Slide 19: Trigonometric Identities

Pythagorean Identities
- sin^2(x) + cos^2(x) = 1
- 1 + tan^2(x) = sec^2(x)
- 1 + cot^2(x) = cosec^2(x)
Reciprocal Identities
Quotient and Co-Quotient Identities
Co-Function Identities

Slide 20: Trigonometric Functions

Definition of Trigonometric Functions
Domain and Range of Trigonometric Functions
Graphs of Trigonometric Functions
Periodicity of Trigonometric Functions
Applications of Trigonometry

Slide 21: Probability - Problem 6

Example Question: There are 20 different balls numbered 1 to 20. A child randomly selects 3 balls without replacement. What is the probability of selecting a ball with a prime number?
Number of favorable outcomes: 8 (2, 3, 5, 7, 11, 13, 17, 19)
Total number of outcomes: 20C3
Probability of selecting a ball with a prime number: P(Prime number) = 8/(20C3)

Slide 22: Probability - Problem 7

A deck of 52 playing cards is shuffled. What is the probability of drawing a heart or a club?
Number of favorable outcomes for heart cards: 13
Number of favorable outcomes for club cards: 13
Number of favorable outcomes for heart or club cards: 26
Total number of outcomes: 52
Probability of drawing a heart or a club: P(Heart or Club) = 26/52 = 1/2

Slide 23: Probability - Problem 8

A bag contains 6 red balls, 4 blue balls, and 5 green balls. Two balls are drawn at random without replacement. Find the probability that both balls are blue.
Sample Space: {R1, R2, R3, R4, R5, R6, B1, B2, B3, B4, G1, G2, G3, G4, G5}
Number of favorable outcomes: 4C2 = 6 (B1B2, B1B3, B1B4, B2B3, B2B4, B3B4)
Total number of outcomes: 15C2 = 105
Probability of both balls being blue: P(2 Blue Balls) = 6/105 = 2/35

Slide 24: Probability - Problem 9

A box contains 8 red balls and 4 green balls. Three balls are drawn at random with replacement. Find the probability that all balls are green.
Sample Space: {R1, R2, R3, R4, R5, R6, R7, R8, G1, G2, G3, G4}
Number of favorable outcomes: 4C3 = 4 (G1G2G3, G1G2G4, G1G3G4, G2G3G4)
Total number of outcomes: 12^3 = 1728
Probability of all balls being green: P(3 Green Balls) = 4/1728 = 1/432

Slide 25: Trigonometry - Problem 1

Find the value of sin(45 degrees) by evaluating the trigonometric ratio.
Sine ratio: sin(45 degrees) = Opposite/Hypotenuse
In a 45-45-90 right-angled triangle, the opposite side and the hypotenuse are equal.
Let the length of both sides be ‘a’, then a/a = 1.
Therefore, sin(45 degrees) = 1

Slide 26: Trigonometry - Problem 2

Find the value of cos(60 degrees) by evaluating the trigonometric ratio.
Cosine ratio: cos(60 degrees) = Adjacent/Hypotenuse
In a 30-60-90 right-angled triangle, the adjacent side is half the length of the hypotenuse.
Let the length of the hypotenuse be ‘a’, then the length of the adjacent side is a/2.
Therefore, cos(60 degrees) = a/2 = 1/2

Slide 27: Trigonometry - Problem 3

Find the value of tan(30 degrees) by evaluating the trigonometric ratio.
Tangent ratio: tan(30 degrees) = Opposite/Adjacent
In a 30-60-90 right-angled triangle, the opposite side is half the length of the adjacent side.
Let the length of the adjacent side be ‘a’, then the length of the opposite side is a/2.
Therefore, tan(30 degrees) = (a/2) / a = 1/2

Slide 28: Trigonometry - Problem 4

Find the value of sec(45 degrees) by evaluating the trigonometric ratio.
Secant ratio: sec(45 degrees) = 1/cos(45 degrees)
We already found that cos(45 degrees) = 1/√2
Therefore, sec(45 degrees) = 1 / (1/√2) = √2

Slide 29: Trigonometry - Problem 5

Find the value of cot(60 degrees) by evaluating the trigonometric ratio.
Cotangent ratio: cot(60 degrees) = 1/tan(60 degrees)
We already found that tan(60 degrees) = √3
Therefore, cot(60 degrees) = 1 / (√3) = 1/√3

Slide 30: Trigonometry - Problem 6

Advance trigonometry problem
- Example Question: A ladder is leaning against a wall. The ladder is 10 meters long and forms an angle of 60 degrees with the ground. How far is the base of the ladder from the wall?
- The base of the ladder, the height of the wall, and the ladder form a right-angled triangle.
- We can use the trigonometric ratio cosine to find the distance.
- Cosine ratio: cos(60 degrees) = Adjacent/Hypotenuse
- Let the distance between the base of the ladder and the wall be ‘x’.
- Therefore, x/10 = cos(60 degrees)
- Solving for x, x = 10 * cos(60 degrees) = 5 meters
- The base of the ladder is 5 meters away from the wall.