Probability - Sample space

  • Definition of probability
  • Sample space of an experiment
  • Events and outcomes
  • Methods of representing a sample space
  • Types of events: certain, impossible, simple, compound
  • Example 1: Tossing a coin
  • Example 2: Rolling a die
  • Example 3: Selecting a card from a deck
  • Probability of an event
  • Law of total probability

Probability - Mutually exclusive events

  • Mutually exclusive events definition
  • Intersection of events
  • Union of events
  • Disjoint events
  • Example 1: Drawing a red or black card
  • Example 2: Tossing a fair coin
  • Example 3: Rolling a die and getting an odd or even number
  • Calculation of probabilities for mutually exclusive events
  • Addition law of probability for mutually exclusive events
  • Notation for mutually exclusive events

Probability - Independent events

  • Independent events definition
  • Joint probability
  • Conditional probability
  • Multiplication law of probability for independent events
  • Example 1: Tossing two fair coins
  • Example 2: Drawing two cards from a deck without replacement
  • Example 3: Rolling two dice and getting a sum of 7
  • Calculation of probabilities for independent events
  • Notation for independent events

Probability - Dependent events

  • Dependent events definition
  • Conditional probability with dependent events
  • Example 1: Drawing two cards from a deck with replacement
  • Example 2: Selecting balls from an urn with replacement
  • Example 3: Drawing two marbles from a bag without replacement
  • Calculation of probabilities for dependent events
  • Multiplication law of probability for dependent events

Probability - Complementary events

  • Complementary events definition
  • Complement of an event
  • Properties of complementary events
  • Example 1: Tossing a fair coin
  • Example 2: Drawing a card from a deck
  • Example 3: Rolling a die
  • Calculation of probabilities for complementary events
  • Notation for complements

Probability - Addition law of probability

  • Addition law of probability definition
  • Union of events
  • Example 1: Tossing a fair coin
  • Example 2: Rolling a die
  • Example 3: Drawing a card from a deck
  • Calculation of probabilities using addition law
  • Notation for addition law of probability

Probability - Multiplication law of probability

  • Multiplication law of probability definition
  • Intersection of events
  • Example 1: Tossing two fair coins
  • Example 2: Drawing two cards from a deck with replacement
  • Example 3: Selecting balls from an urn
  • Calculation of probabilities using multiplication law
  • Notation for multiplication law of probability

Probability - Conditional probability

  • Conditional probability definition
  • Example 1: Drawing a card of a particular suit
  • Example 2: Selecting balls from an urn
  • Example 3: Rolling dice and getting a sum divisible by 3
  • Calculation of conditional probabilities
  • Notation for conditional probability

Probability - Bayes’ theorem

  • Bayes’ theorem definition
  • Theorem statement
  • Example 1: Cancer screening test
  • Example 2: Selecting a card from a deck
  • Example 3: Tossing three fair coins
  • Calculation of probabilities using Bayes’ theorem
  • Applications of Bayes’ theorem
  1. Probability - Sample space
  • Definition of probability
  • Sample space of an experiment
  • Events and outcomes
  • Methods of representing a sample space
  • Types of events: certain, impossible, simple, compound
  1. Example 1: Tossing a coin
  • Sample space: {H, T}
  • Events: {H}, {T}, {H, T}
  • Probability of getting heads: P(H) = 1/2
  • Probability of getting tails: P(T) = 1/2
  1. Example 2: Rolling a die
  • Sample space: {1, 2, 3, 4, 5, 6}
  • Events: {1}, {2}, {3}, {4}, {5}, {6}
  • Probability of getting a particular number, e.g., P(3) = 1/6
  1. Example 3: Selecting a card from a deck
  • Sample space: 52 cards in a standard deck
  • Events: drawing a certain card, drawing a red card, drawing a face card
  • Calculating probabilities for different events using the size of the sample space and the number of favorable outcomes
  1. Probability of an event
  • Probability of an event A: P(A)
  • Calculating probability using the formula: P(A) = favorable outcomes / total outcomes
  • Properties of probability: 0 <= P(A) <= 1
  • Probability of an impossible event: P(A) = 0
  • Probability of a certain event: P(A) = 1
  1. Law of total probability
  • Law of total probability formula: P(A) = P(A ∩ B1) + P(A ∩ B2) + … + P(A ∩ Bn)
  • Applying the law of total probability to calculate the probability of an event A when the sample space is partitioned into mutually exclusive events B1, B2, …, Bn
  1. Probability - Mutually exclusive events
  • Mutually exclusive events definition
  • Intersection of events
  • Union of events
  • Disjoint events
  • Example 1: Drawing a red or black card
  • Example 2: Tossing a fair coin
  1. Example 3: Rolling a die and getting an odd or even number
  • Calculating probabilities for mutually exclusive events using the addition law of probability
  • P(A ∪ B) = P(A) + P(B) when A and B are mutually exclusive events
  1. Probability - Independent events
  • Independent events definition
  • Joint probability
  • Conditional probability
  • Multiplication law of probability for independent events
  • Example 1: Tossing two fair coins
  • Example 2: Drawing two cards from a deck without replacement
  1. Example 3: Rolling two dice and getting a sum of 7
  • Calculating probabilities for independent events using the multiplication law of probability
  • P(A ∩ B) = P(A) * P(B) when A and B are independent events
  1. Types of events: certain, impossible, simple, compound
  • Certain events: Events that will always occur, with a probability of 1.
  • Impossible events: Events that will never occur, with a probability of 0.
  • Simple events: Events that consist of a single outcome. For example, getting a specific number when rolling a die.
  • Compound events: Events that consist of more than one outcome. For example, getting an even number when rolling a die.
  1. Example 1: Tossing a fair coin
  • Sample space: {H, T}
  • Possible events: {H}, {T}, {H, T}
  • Certain event: Getting either heads or tails (H or T)
  • Impossible event: Getting neither heads nor tails (not H and not T)
  1. Example 2: Rolling a die
  • Sample space: {1, 2, 3, 4, 5, 6}
  • Possible events: {1}, {2}, {3}, {4}, {5}, {6}
  • Certain event: Getting any number from 1 to 6
  • Impossible event: Getting a number less than 1 or greater than 6
  1. Example 3: Selecting a card from a deck
  • Sample space: 52 cards in a standard deck
  • Possible events: Drawing a specific card, drawing a red card, drawing a face card
  • Certain event: Drawing any card from the deck
  • Impossible event: Drawing a card that is not in the deck
  1. Calculation of probabilities for different events
  • Probability of an event A: P(A)
  • Calculating probability using the formula: P(A) = favorable outcomes / total outcomes
  • For example, the probability of drawing a face card from a deck is P(face card) = 12 / 52 = 3 / 13
  1. Law of total probability
  • Law of total probability formula: P(A) = P(A ∩ B1) + P(A ∩ B2) + … + P(A ∩ Bn)
  • Applying the law of total probability to calculate the probability of an event A when the sample space is partitioned into mutually exclusive events B1, B2, …, Bn
  • For example, if we partition the sample space of rolling a fair die into the events of getting an even or odd number, the probability of getting a 3 can be calculated using P(3) = P(3 and even) + P(3 and odd)
  1. Example 1: Tossing two fair coins
  • Sample space: {HH, HT, TH, TT}
  • Possible events: {HH}, {HT}, {TH}, {TT}
  • Certain event: Getting any combination of heads and tails
  • Impossible event: Getting a combination that is not in the sample space
  1. Example 2: Drawing two cards from a deck with replacement
  • Sample space: 52 cards in a standard deck
  • Possible events: Drawing a specific pair of cards, drawing two red cards, drawing two spades
  • Certain event: Drawing any two cards from the deck
  • Impossible event: Drawing two cards that are not in the deck
  1. Example 3: Selecting balls from an urn
  • Sample space: Number of balls in the urn
  • Possible events: Drawing a specific ball, drawing a red ball, drawing an even-numbered ball
  • Certain event: Drawing any ball from the urn
  • Impossible event: Drawing a ball that is not in the urn
  1. Calculation of probabilities for dependent events
  • Probabilities for dependent events can be calculated using the multiplication law of probability
  • P(A ∩ B) = P(A) * P(B|A)
  • For example, if we want to calculate the probability of drawing two red cards without replacement, we first determine the probability of drawing a red card on the first draw, followed by the probability of drawing a red card on the second draw given that a red card was already drawn on the first draw.