Probability

  • Definition of probability
  • Types of probability
  • Sample space and events
  • Random experiments
  • Equally likely outcomes

Definition of Probability

  • Probability is the measure of the likelihood that an event will occur.
  • It is denoted by P(A) where A is the event.
  • Probability values range from 0 to 1.
  • An event with probability 0 is impossible, while an event with probability 1 is certain.

Types of Probability

  1. Theoretical Probability:
    • Based on the theoretical analysis of events.
    • Calculated by dividing the number of favorable outcomes by the total number of possible outcomes.
  1. Experimental Probability:
    • Based on experimental results from repeated trials.
    • Calculated by dividing the number of times the event occurs by the total number of trials.

Sample Space and Events

  • Sample Space: The collection of all possible outcomes of a random experiment, denoted by S.
  • Event: A subset of the sample space, denoted by A.
  • An event may consist of one or more sample points. Example:
  • In tossing a fair coin, the sample space is {H, T}, where H represents heads and T represents tails.
  • Event A can be the event of getting heads, which is {H}.

Random Experiments

  • Random Experiment: An experiment whose outcome cannot be predicted with certainty.
  • Examples of random experiments:
    • Tossing a coin
    • Rolling a dice
    • Drawing a card from a deck
  • Each trial of a random experiment is independent and has multiple possible outcomes.

Equally Likely Outcomes

  • Equally Likely Outcomes: Outcomes of an experiment that have equal chances of occurring.
  • If all outcomes are equally likely, the probability of an event A occurring is given by:
    • P(A) = Number of outcomes favorable to event A / Total number of outcomes
  • Equally likely outcomes help calculate probabilities easily. Example:
  • In rolling a fair six-sided die, each outcome has a probability of 1/6.

Union and Intersection of Events

  • Union (A ∪ B): The event that occurs if either A or B or both occur.
  • Intersection (A ∩ B): The event that occurs only if both A and B occur.
  • Complement (A’) or (AC): The event that occurs when A does not occur. Example:
  • Tossing a fair coin:
    • A: Getting heads
    • B: Getting tails
    • A ∪ B: Getting either heads or tails
    • A ∩ B: Impossible event (cannot get both heads and tails simultaneously)

Addition and Multiplication Rules

  • Addition Rule:
    • P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
    • Formula to calculate the probability of the union of two events.
  • Multiplication Rule:
    • P(A ∩ B) = P(A) * P(B | A)
    • Formula to calculate the probability of the intersection of two events. Note: P(B | A) represents the probability of event B given that event A has occurred.

Conditional Probability

  • Conditional Probability: Probability of an event B occurring given that event A has already occurred, denoted as P(B | A).
  • It can be calculated using the formula:
    • P(B | A) = P(A ∩ B) / P(A) Example:
  • From a deck of cards:
    • A: Getting a red card
    • B: Getting a heart
    • P(B | A): Probability of getting a heart given that a red card has already been drawn.

Sure! Below are slides 11 to 20 on the topic of Probability for the 12th Boards exam:

  1. Conditional Probability (continued)
  • P(B | A) = P(A ∩ B) / P(A)
  • Examples of conditional probability:
    • A: Selecting a black card from a standard deck of cards
    • B: Selecting a spade
      • P(B | A): Probability of selecting a spade given that a black card has already been drawn
  • Conditional probability helps analyze the dependence between events.
  1. Laws of Probability
  • Addition Law of Probability:
    • P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
    • Formula to calculate the probability of the union of two events.
  • Multiplication Law of Probability:
    • P(A ∩ B) = P(A) * P(B | A)
    • Formula to calculate the probability of the intersection of two events.
  • These laws are fundamental to solving probability problems.
  1. Independent Events
  • Independent events: Two events A and B are independent if the occurrence or non-occurrence of one event does not affect the occurrence or non-occurrence of the other.
    • P(A ∩ B) = P(A) * P(B)
  • Example:
    • A: Tossing a fair coin and getting heads
    • B: Rolling a fair die and getting a 4
    • In this case, the outcome of the coin toss does not affect the outcome of the die roll.
  1. Dependent Events
  • Dependent events: Two events A and B are dependent if the occurrence or non-occurrence of one event affects the occurrence or non-occurrence of the other.
    • P(A ∩ B) = P(A) * P(B | A)
  • Example:
    • A: Drawing a red ball from a bag with 3 red balls and 2 blue balls
    • B: Drawing a blue ball after replacing the first ball
    • In this case, the second event depends on the outcome of the first event.
  1. Mutually Exclusive Events
  • Mutually exclusive events: Two events A and B are mutually exclusive if they cannot occur at the same time.
    • P(A ∩ B) = 0
  • Example:
    • A: Rolling a die and getting a 1
    • B: Rolling a die and getting a 6
    • It is impossible to get both a 1 and a 6 on a single roll of a die.
  1. Not Mutually Exclusive Events
  • Not mutually exclusive events: Two events A and B are not mutually exclusive if they can occur at the same time.
    • P(A ∩ B) ≠ 0
  • Example:
    • A: Tossing a fair coin and getting heads
    • B: Tossing a fair coin and getting tails
    • These events can occur simultaneously, as there are two possible outcomes.
  1. Complementary Events
  • Complementary events: The complement of an event A, denoted by A’ or AC, is the event that occurs when A does not occur.
    • P(A’) = 1 - P(A)
  • Example:
    • A: Drawing a red card from a standard deck of cards
    • A’: Drawing a non-red card (black)
    • The probability of drawing a non-red card is 1 minus the probability of drawing a red card.
  1. Probability Distribution
  • Probability distribution: A table or an equation that describes the probabilities of various outcomes of a random experiment.
  • Discrete probability distribution:
    • Only a countable number of outcomes.
    • Each outcome has a non-negative probability.
  • Continuous probability distribution:
    • Infinitely many outcomes within a range.
    • Probability is described by a probability density function.
  1. Expected Value
  • Expected value: The mean or the average value of a random variable, denoted by E(X).
  • It represents the long-term average or the ’expected’ value over many repeated trials.
  • For a discrete random variable:
    • E(X) = Σ(x * P(X=x))
  • For a continuous random variable:
    • E(X) = ∫(x * f(x)) dx
  • The expected value helps in analyzing the outcome of random experiments.
  1. Variance and Standard Deviation
  • Variance: A measure that describes how spread out the values of a random variable are around the mean, denoted by Var(X).
  • Standard Deviation: The square root of the variance, denoted by σ(X) or SD(X).
  • For a discrete random variable:
    • Var(X) = E((X - E(X))^2) = Σ((x - μ)^2 * P(X=x))
  • For a continuous random variable:
    • Var(X) = E((X - E(X))^2) = ∫((x - μ)^2 * f(x)) dx
  • Variance and standard deviation help in quantifying the variability of random variables.

That concludes slides 11 to 20 on the topic of Probability for the 12th Boards Math exam.

Probability - Example of Formula

  • Example of Theoretical Probability:
    • A: Drawing a red ball from a bag with 3 red balls and 2 blue balls
      • P(A) = Number of red balls / Total number of balls
  • Example of Experimental Probability:
    • A: Rolling a fair die and getting an even number
      • P(A) = Number of times an even number is rolled / Total number of rolls

Probability of Simple and Compound Events

  • Simple Event: An event with only one outcome
    • Example: Rolling a die and getting a 3
  • Compound Event: An event with more than one outcome
    • Example: Rolling a die and getting an even number (2, 4, or 6)
  • Probability of a simple event can be calculated directly, while the probability of a compound event requires additional calculations.

Probability of Independent Events

  • Independent events do not affect each other’s likelihood of occurring.
  • If A and B are independent events:
    • P(A ∩ B) = P(A) * P(B)
  • Example: Tossing a fair coin twice:
    • A: Getting heads on the first toss
    • B: Getting tails on the second toss
    • P(A) = P(B) = 1/2
    • P(A ∩ B) = P(A) * P(B) = 1/4

Probability of Dependent Events

  • Dependent events are influenced by each other’s occurrence.
  • If A and B are dependent events:
    • P(A ∩ B) = P(A) * P(B | A)
  • Example: Drawing cards from a standard deck without replacement:
    • A: Drawing a red card
    • B: Drawing a second red card
    • P(A) = 26/52
    • P(B | A) = 25/51
    • P(A ∩ B) = P(A) * P(B | A) = 25/102

Probability of Mutually Exclusive Events

  • Mutually exclusive events cannot occur simultaneously.
  • If A and B are mutually exclusive events:
    • P(A ∩ B) = 0
  • Example: Rolling a fair die:
    • A: Getting a 2
    • B: Getting a 5
    • P(A) = 1/6
    • P(B) = 1/6
    • P(A ∩ B) = 0

Conditional Probability using a Table

  • A probability table helps calculate conditional probabilities.
  • Example: Drawing a card from a standard deck:
    • A: Drawing a red card
    • B: Drawing a heart
    • Probability table shows the number of favorable outcomes for each event.
  • P(B | A) = Number of outcomes favorable to both A and B / Number of outcomes favorable to A

Expected Value and Fair Games

  • Expected value is the average value of a random variable over many trials.
  • For a discrete random variable X:
    • E(X) = Σ(x * P(X=x))
    • Represents the long-term average value of X.
  • Fair game: A game with an expected value of zero.
    • Example: Tossing a fair coin for $1:
      • P(Win $1) = P(Lose $1) = 1/2
      • E(Winnings) = (1 * 1/2) + (-1 * 1/2) = $0

Variance and Standard Deviation

  • Variance measures the spread of a random variable’s values around the mean.
  • Standard deviation is the square root of the variance.
  • For a discrete random variable X with mean μ:
    • Var(X) = E((X - μ)^2) = Σ((x - μ)^2 * P(X=x))
    • SD(X) = √Var(X)
  • These measures help quantify the variability and dispersion of random variable values.

Combination and Permutation

  • Combination: The number of ways to choose k items from a group of n, irrespective of their order.
    • nCr = n! / (r! * (n-r)!)
    • Example: Choosing 2 students from a class of 10 for a group project.
  • Permutation: The number of ways to arrange items in a specific order.
    • nPr = n! / (n-r)!
    • Example: Arranging 3 books on a shelf out of a collection of 6.

That concludes slides 21 to 30 on the topic of Probability for the 12th Boards Math exam.