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Probability: Rules & Distributions

102 cards · 🇺🇸 English

Probability rules, independence, conditional probability and common distributions.

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Probability

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  1. 1Probability
    A numerical measure between 0 and 1 describing how likely an event is to occur
  2. 2a probability of 0
    The event is impossible (or almost surely will not occur)
  3. 3a probability of 1
    The event is certain (or almost surely will occur)
  4. 4Sample space
    The set of all possible outcomes of an experiment, often denoted Ω or S
  5. 5Event in probability
    A subset of the sample space; a collection of possible outcomes
  6. 6Complement of event A
    The event 'not A', denoted A' or Ā; P(A') = 1 − P(A)
  7. 7Probability of getting heads on a fair coin
    1/2 or 0.5 or 50% (two equally likely outcomes)
  8. 8Probability of rolling a 4 on a fair six-sided die
    1/6 (one desired outcome out of six equally likely outcomes)
  9. 9Addition rule for mutually exclusive events
    P(A or B) = P(A) + P(B) when A and B cannot occur simultaneously
  10. 10Mutually exclusive events
    Events that cannot occur at the same time; P(A and B) = 0
  11. 11General addition rule for any two events
    P(A or B) = P(A) + P(B) − P(A and B)
  12. 12Why subtract P(A and B) in the general addition rule?
    To avoid double-counting outcomes that are in both A and B
  13. 13Independent events
    Events where the occurrence of one does not affect the probability of the other
  14. 14Multiplication rule for independent events
    P(A and B) = P(A) × P(B) when A and B are independent
  15. 15Probability of flipping two heads in a row
    1/2 × 1/2 = 1/4 (independent events, multiply probabilities)
  16. 16Conditional probability
    The probability of event A given that event B has occurred, denoted P(A|B)
  17. 17Formula for conditional probability
    P(A|B) = P(A and B) / P(B), assuming P(B) > 0
  18. 18General multiplication rule
    P(A and B) = P(A|B) × P(B) = P(B|A) × P(A)
  19. 19Bayes' theorem
    P(A|B) = [P(B|A) × P(A)] / P(B), relating prior and posterior probabilities
  20. 20Law of total probability
    P(B) = Σ P(B|Aᵢ) × P(Aᵢ), where {Aᵢ} partitions the sample space
  21. 21Events to be exhaustive
    Together they cover all possible outcomes (union equals the sample space)
  22. 22Probability distribution
    A function that describes the probabilities of all possible outcomes of a random variable
  23. 23Discrete random variable
    A variable that can take on a countable number of distinct values
  24. 24Continuous random variable
    A variable that can take on any value in a continuous range (uncountable)
  25. 25Probability mass function (PMF)
    The probability distribution of a discrete random variable, giving P(X = x)
  26. 26Probability density function (PDF)
    The function describing the probability distribution of a continuous random variable
  27. 27Expected value (mean) of a discrete random variable
    E(X) = Σ xᵢ · P(X = xᵢ), the weighted average of all possible values
  28. 28Expected value for a continuous random variable
    E(X) = ∫ x · f(x) dx, where f(x) is the probability density function
  29. 29Variance of a random variable
    Var(X) = E[(X − μ)²] = E(X²) − [E(X)]², measuring spread around the mean
  30. 30Bernoulli distribution
    A discrete distribution with two outcomes: success (p) and failure (1−p)

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