Probability of an Event

  • Probability is a measure of the likelihood of an event occurring.
  • It is expressed as a number between 0 and 1, with 0 representing impossibility and 1 representing certainty.
  • The probability of an event can be determined using various methods.
  • The probability of an event A is denoted as P(A).
  • P(A) is calculated as the ratio of the number of favorable outcomes to the total number of possible outcomes.
  • P(A) = Number of favorable outcomes / Total number of possible outcomes

Example 1:

  • Consider rolling a fair six-sided die.
  • What is the probability of getting an even number?
  • Favorable outcomes: {2, 4, 6}
  • Total outcomes: {1, 2, 3, 4, 5, 6}
  • P(even number) = 3/6 = 1/2 = 0.5
  • The probability of an event can also be represented as a fraction, decimal, or percentage.
  • Fractions, decimals, and percentages are different ways of expressing the same value.

Example 2:

  • The probability of getting a head when flipping a fair coin is 1/2.
  • This can also be represented as a decimal: 0.5 or a percentage: 50%.
  • If an event is impossible, its probability is 0.
  • If an event is certain, its probability is 1.
  • The probability of any event lies between 0 and 1.
  • The complement of an event A is the event that A does not occur.
  • The probability of the complement of A, denoted as P(A’), is equal to 1 minus the probability of A.
  • P(A’) = 1 - P(A)

Example 3:

  • Consider rolling a fair six-sided die.
  • What is the probability of not rolling a 3?
  • Favorable outcomes: {1, 2, 4, 5, 6}
  • Total outcomes: {1, 2, 3, 4, 5, 6}
  • P(not rolling a 3) = 5/6
  • In some cases, events can be combined to form new events.
  • The probability of the union of two events A and B, denoted as P(A ∪ B), is the probability that either A or B or both occur.
  • P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

Example 4:

  • Consider drawing a card from a standard deck of 52 cards.
  • What is the probability of drawing a heart or a queen?
  • P(heart or queen) = P(heart) + P(queen) - P(heart and queen)
  • The probability of the intersection of two events A and B, denoted as P(A ∩ B), is the probability that both A and B occur.
  • P(A ∩ B) = P(A) * P(B|A)

Example 5:

  • Consider drawing two cards from a standard deck of 52 cards, without replacement.
  • What is the probability of drawing two aces?
  • P(drawing two aces) = P(drawing an ace first) * P(drawing an ace second, given that an ace was already drawn)

This is the end of the first 10 slides.

  1. Conditional Probability
  • Conditional probability is the probability of an event occurring given that another event has already occurred.
  • It is denoted as P(A|B), which means “the probability of event A given that event B has occurred.”
  • P(A|B) = P(A and B) / P(B)
  • Example: Consider flipping two fair coins. What is the probability of getting a head on the second coin given that the first coin is a head?
    • P(H2|H1) = P(H1 and H2) / P(H1)
    • Favorable outcome for H1 and H2 = {(H, H)}
    • P(H2|H1) = 1/2
  1. Independent Events
  • Two events A and B are independent if the occurrence of one event does not affect the occurrence of the other event.
  • The probability of two independent events A and B occurring together is the product of their individual probabilities.
  • P(A and B) = P(A) * P(B)
  • Example: Consider rolling two fair six-sided dice. What is the probability of getting a 4 on the first die and a 3 on the second die?
    • P(4 on 1st die and 3 on 2nd die) = P(4 on 1st die) * P(3 on 2nd die)
    • P(4 on 1st die and 3 on 2nd die) = 1/6 * 1/6 = 1/36
  1. Mutually Exclusive Events
  • Mutually exclusive events are events that cannot occur at the same time.
  • If two events A and B are mutually exclusive, then the probability of either A or B occurring is the sum of their individual probabilities.
  • P(A or B) = P(A) + P(B)
  • Example: Consider rolling a fair six-sided die. What is the probability of getting a 2 or a 5?
    • P(2 or 5) = P(2) + P(5) = 1/6 + 1/6 = 1/3
  1. Addition Rule for Probability
  • The addition rule for probability states that the probability of the union of two events is equal to the sum of their individual probabilities minus the probability of their intersection.
  • P(A or B) = P(A) + P(B) - P(A and B)
  • Example: Consider drawing a card from a standard deck of 52 cards. What is the probability of drawing a heart or a king?
    • P(heart or king) = P(heart) + P(king) - P(heart and king)
    • P(heart or king) = 13/52 + 4/52 - 1/52 = 16/52 = 4/13
  1. Multiplication Rule for Probability
  • The multiplication rule for probability states that the probability of the intersection of two independent events is equal to the product of their individual probabilities.
  • P(A and B) = P(A) * P(B)
  • Example: Consider flipping two fair coins. What is the probability of getting a head on both coins?
    • P(H on 1st coin and H on 2nd coin) = P(H on 1st coin) * P(H on 2nd coin)
    • P(H on 1st coin and H on 2nd coin) = 1/2 * 1/2 = 1/4
  1. Factorial Notation
  • Factorial notation is used to represent the number of ways a set of distinct objects can be arranged.
  • The factorial of a positive integer n is denoted as n!, and it is the product of all positive integers less than or equal to n.
  • n! = n * (n-1) * (n-2) * … * 3 * 2 * 1
  • Example: 5! = 5 * 4 * 3 * 2 * 1 = 120
  1. Permutations
  • Permutations are arrangements of a set of objects in a specific order.
  • The number of permutations of n distinct objects taken r at a time is given by the formula:
  • P(n, r) = n! / (n - r)!
  • Example: How many ways can 3 different items be arranged in a line?
    • P(3, 3) = 3! / (3 - 3)! = 3! / 0! = 3
  1. Combinations
  • Combinations are selections of objects from a set without regard to order.
  • The number of combinations of n distinct objects taken r at a time is given by the formula:
  • C(n, r) = n! / (r! * (n - r)!)
  • Example: How many different groups of 2 can be formed from a set of 5 people?
    • C(5, 2) = 5! / (2! * (5 - 2)!) = 5! / (2! * 3!) = 5 * 4 / (2 * 1) = 10
  1. Expected Value
  • The expected value of a random variable is a measure of its average value over many trials.
  • It is calculated by multiplying each possible outcome by its probability and summing all the products.
  • E(X) = x1 * P(x1) + x2 * P(x2) + … + xn * P(xn)
  • Example: What is the expected value of rolling a fair six-sided die?
    • E(X) = 1 * 1/6 + 2 * 1/6 + 3 * 1/6 + 4 * 1/6 + 5 * 1/6 + 6 * 1/6 = 3.5
  1. Variance and Standard Deviation
  • Variance and standard deviation are measures of the spread or dispersion of a random variable’s values around its expected value.
  • Variance is calculated by taking the average of the squared differences between each value and the expected value.
  • Standard deviation is the square root of the variance.
  • Var(X) = (x1 - E(X))^2 * P(x1) + (x2 - E(X))^2 * P(x2) + … + (xn - E(X))^2 * P(xn)
  • SD(X) = √Var(X)
  • Example: What is the variance and standard deviation of rolling a fair six-sided die?
    • Var(X) = (1 - 3.5)^2 * 1/6 + (2 - 3.5)^2 * 1/6 + … + (6 - 3.5)^2 * 1/6
    • SD(X) = √Var(X)

Probability - Conditional Probability

  1. Conditional Probability
  • Conditional probability is the probability of an event occurring given that another event has already occurred.
  • It is denoted as P(A|B), which means “the probability of event A given that event B has occurred.”
  • P(A|B) = P(A and B) / P(B)
  1. Example: Flipping Two Coins
  • Consider flipping two fair coins.
  • What is the probability of getting a head on the second coin given that the first coin is a head?
  • Favorable outcome for H1 and H2 = {(H, H)}
  • P(H2|H1) = P(H1 and H2) / P(H1)
  • P(H2|H1) = 1/2
  1. Example: Drawing Cards
  • Consider drawing a card from a standard deck of 52 cards.
  • What is the probability of drawing a red card given that the card drawn is a heart?
  • Favorable outcomes for red card and heart = {H, D}
  • P(red card|heart) = P(red card and heart) / P(heart)
  • P(red card|heart) = 26/52 / 13/52 = 26/13 = 2
  1. Independent Events
  • Two events A and B are independent if the occurrence of one event does not affect the occurrence of the other event.
  • The probability of two independent events A and B occurring together is the product of their individual probabilities.
  • P(A and B) = P(A) * P(B)
  1. Example: Rolling Two Dice
  • Consider rolling two fair six-sided dice.
  • What is the probability of getting a 4 on the first die and a 3 on the second die?
  • P(4 on 1st die and 3 on 2nd die) = P(4 on 1st die) * P(3 on 2nd die)
  • P(4 on 1st die and 3 on 2nd die) = 1/6 * 1/6 = 1/36
  1. Mutually Exclusive Events
  • Mutually exclusive events are events that cannot occur at the same time.
  • If two events A and B are mutually exclusive, then the probability of either A or B occurring is the sum of their individual probabilities.
  • P(A or B) = P(A) + P(B)
  1. Example: Drawing Cards
  • Consider drawing a card from a standard deck of 52 cards.
  • What is the probability of drawing a heart or a spade?
  • P(heart or spade) = P(heart) + P(spade) = 13/52 + 13/52 = 26/52 = 1/2
  1. Example: Rolling a Die
  • Consider rolling a fair six-sided die.
  • What is the probability of rolling a 2 or a 3?
  • P(2 or 3) = P(2) + P(3) = 1/6 + 1/6 = 1/3
  1. Addition Rule for Probability
  • The addition rule for probability states that the probability of the union of two events is equal to the sum of their individual probabilities minus the probability of their intersection.
  • P(A or B) = P(A) + P(B) - P(A and B)
  1. Example: Drawing Cards
  • Consider drawing a card from a standard deck of 52 cards.
  • What is the probability of drawing a heart or a king?
  • P(heart or king) = P(heart) + P(king) - P(heart and king)
  • P(heart or king) = 13/52 + 4/52 - 1/52 = 16/52 = 4/13