Probability - Examples and properties of probability

  • Probability is a branch of mathematics that deals with the study of uncertainty and randomness.
  • It is used to describe the likelihood of an event occurring.
  • The probability of an event is a number between 0 and 1, inclusive.
  • The sum of the probabilities of all possible outcomes of an experiment is always 1.
  • Probability can be expressed as fractions, decimals, or percentages.

Basic Definitions

  • Sample Space: The set of all possible outcomes of an experiment, usually denoted by S.
  • Event: Any subset of the sample space.
  • Probability of an Event: The likelihood that a specific event will occur, denoted by P(event).
  • Null Event: The event that has no outcomes, denoted by ∅.

Properties of Probability

  1. Addition Rule: The probability of the union of two events is given by the sum of their individual probabilities.
    • P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
  1. Multiplication Rule: The probability of the intersection of two independent events is given by the product of their individual probabilities.
    • P(A ∩ B) = P(A) * P(B)
  1. Complement Rule: The probability of an event not occurring is given by subtracting the probability of the event from 1.
    • P(A’) = 1 - P(A)
  1. Multiplication Rule for Dependent Events: The probability of the intersection of two dependent events is given by the product of their individual probabilities, given that the first event has already occurred.
    • P(A ∩ B) = P(A) * P(B|A)

Examples of Probability

  1. Tossing a fair coin:
    • Sample Space: {H, T}
    • P(H) = 1/2
    • P(T) = 1/2
  1. Rolling a fair die:
    • Sample Space: {1, 2, 3, 4, 5, 6}
    • P(odd number) = 3/6 = 1/2
    • P(even number) = 3/6 = 1/2
  1. Drawing a card from a standard deck:
    • Sample Space: 52 cards
    • P(hearts) = 13/52 = 1/4
    • P(diamonds) = 13/52 = 1/4

More Examples of Probability

  1. Selecting a red ball from a bag containing red and blue balls:
    • Sample Space: {red, blue}
    • P(red) = 1/2
    • P(blue) = 1/2
  1. Flipping two coins:
    • Sample Space: {(H, H), (H, T), (T, H), (T, T)}
    • P(at least one head) = 3/4
    • P(two tails) = 1/4
  1. Drawing two cards without replacement:
    • Sample Space: 52 cards
    • P(king, then queen) = (4/52) * (4/51)

Conditional Probability

  • Conditional probability is the probability of an event occurring given that another event has already occurred.
  • It is denoted by P(A|B), read as “the probability of A given B”.

That concludes our introduction to examples and properties of probability. Remember to practice using these concepts to solve various problems.

Probability - Examples and properties of probability

Slide 11

  • Conditional probability is the probability of an event occurring given that another event has already occurred.
  • It is denoted by P(A|B), read as “the probability of A given B”.
  • Conditional probability is calculated using the formula:
    • P(A|B) = P(A ∩ B) / P(B)

Example: Drawing cards from a deck

  • Suppose we have a standard deck of cards.
  • Let event A be drawing a red card.
  • Let event B be drawing a heart.
  • We want to find the probability of drawing a red card given that the card is a heart.
  • P(A|B) = P(A ∩ B) / P(B)
  • P(A ∩ B) = 1/52
  • P(B) = 1/4
  • P(A|B) = (1/52) / (1/4) = 1/13

Slide 13

  • Two events A and B are said to be independent if the occurrence of one event does not affect the occurrence of the other event.
  • The multiplication rule for independent events is given by:
    • P(A ∩ B) = P(A) * P(B)

Example: Tossing a coin and rolling a die

  • Let event A be getting a head when tossing a fair coin.
  • Let event B be rolling a 3 on a fair die.
  • We want to find the probability of getting a head and rolling a 3.
  • P(A ∩ B) = P(A) * P(B)
  • P(A) = 1/2
  • P(B) = 1/6
  • P(A ∩ B) = (1/2) * (1/6) = 1/12

Slide 15

  • The addition rule for two events A and B is given by:
    • P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

Example: Drawing cards from a deck

  • Suppose we have a standard deck of cards.
  • Let event A be drawing a red card.
  • Let event B be drawing a heart.
  • We want to find the probability of drawing a red card or a heart.
  • P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
  • P(A) = 1/2
  • P(B) = 1/4
  • P(A ∩ B) = 1/52
  • P(A ∪ B) = (1/2) + (1/4) - (1/52) = 27/52

Slide 17

  • The complement rule states that the probability of an event not occurring is given by subtracting the probability of the event from 1.
    • P(A’) = 1 - P(A)

Example: Rolling a die

  • Let event A be rolling an even number on a fair die.
  • We want to find the probability of rolling an odd number.
  • P(A’) = 1 - P(A)
  • P(A) = 3/6 = 1/2
  • P(A’) = 1 - (1/2) = 1/2

Slide 19

  • The multiplication rule for dependent events is given by:
    • P(A ∩ B) = P(A) * P(B|A)
    • P(B|A) is the probability of event B occurring given that event A has already occurred.

Example: Drawing cards from a deck

  • Suppose we have a standard deck of cards.
  • Let event A be drawing a red card.
  • Let event B be drawing a heart, given that a red card has already been drawn.
  • We want to find the probability of drawing a heart after drawing a red card.
  • P(A) = 1/2
  • P(B|A) = 13/51
  • P(A ∩ B) = (1/2) * (13/51) = 13/102

Conditional Probability

  • Conditional probability is the probability of an event occurring given that another event has already occurred.
  • It is denoted by P(A|B), read as “the probability of A given B”.
  • Conditional probability is calculated using the formula:
    • P(A|B) = P(A ∩ B) / P(B)
  • Example:
    • Suppose we have a deck of cards. Let event A be drawing a red card and event B be drawing a heart. If we draw a card and it is a heart, what is the probability that it is also red?

Independent Events

  • Two events A and B are said to be independent if the occurrence of one event does not affect the occurrence of the other event.
  • The multiplication rule for independent events is given by:
    • P(A ∩ B) = P(A) * P(B)
  • Example:
    • Tossing a fair coin and rolling a fair die. What is the probability of getting a head and rolling a 3?

Addition Rule

  • The addition rule for two events A and B is given by:
    • P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
  • Example:
    • Drawing cards from a deck. What is the probability of drawing a red card or a heart?

Complement Rule

  • The complement rule states that the probability of an event not occurring is given by subtracting the probability of the event from 1.
    • P(A’) = 1 - P(A)
  • Example:
    • Rolling a die. What is the probability of not rolling an even number?

Dependent Events

  • The multiplication rule for dependent events is given by:
    • P(A ∩ B) = P(A) * P(B|A)
    • P(B|A) is the probability of event B occurring given that event A has already occurred.
  • Example:
    • Drawing cards from a deck. What is the probability of drawing a heart after drawing a red card?

Bayes’ Theorem

  • Bayes’ theorem is a formula used to find the probability of a particular event given prior knowledge of related events.
  • It is given by:
    • P(A|B) = (P(B|A) * P(A)) / P(B)
  • Example:
    • Suppose we have two bags, one containing 5 red marbles and 3 blue marbles, and the other containing 4 red marbles and 6 blue marbles. If we randomly select a bag and draw a red marble, what is the probability that it came from the first bag?

Permutations

  • Permutations are arrangements of objects in a particular order.
  • The number of permutations of n objects taken r at a time is given by:
    • P(n, r) = n! / (n - r)!
  • Example:
    • How many different ways can 4 students be seated in a row of chairs?

Combinations

  • Combinations are selections of objects without regard to order.
  • The number of combinations of n objects taken r at a time is given by:
    • C(n, r) = n! / (r! * (n - r)!)
  • Example:
    • How many different committees can be formed from a group of 8 people?

Expected Value

  • The expected value of a random variable is a weighted average of its possible values, where the weights are the probabilities of each value occurring.
  • It is given by:
    • E(X) = Σ (x * P(X = x))
  • Example:
    • A fair die is rolled. What is the expected value of the number rolled?

Law of Large Numbers

  • The law of large numbers states that as the number of trials of a random experiment increases, the observed average of the outcomes will approach the expected value.
  • Example:
    • Tossing a fair coin. As the number of coin tosses increases, the proportion of heads observed will approach 0.5.