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Module 2: Probability

Probability is the study of uncertainty and randomness in the world. It measures chance.

Proportion is a summary statistic. Proportion measures size (i.e. how many patients have optional blood pressure).

  • We show the probability of Event A as P(A)
  • The compliment of an event, or chance of event not occurring is AC
    • P(A) + P(AC) = 1
  • Two events are mutually exclusive (or "disjoint") if when one event occurs the other cannot
  • Joint Probability = P(A & B) = P(A,B) = P(A ꓵ B)

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  • P(A or B) = P(A U B)
    • When A and B are non-mutually exclusive P(A or B) = P(A) + P(B) - P(A & B)
    • When A and B are mutually exclusive events P(A or B) = P(A) + P(B)   since P(A&B)=0

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  • Conditional probability is the chance of an event occurring given that another event occurred, writen as P(A|B) = The probability of A given B.
    • P(A|B) = P(A&B)/P(B)
    • P(A|B) * P(B) = P(A&B)

Bayes' Theorem

Because P(A & B) = P(B & A),
P(A & B) = P(B & A) = P(B|A) × P(A)
=
P(A|B) × P(B) = P(B|A) × P(A)

P(B|A) = P(A|B) * P(B) / P(A)

Sensitivity and Conditional Probabilities

Sensitivity of a screening test = Probability of positive test given the person has the disease, this can be expressed as P(X  = +| Y = +) = the probability of a disease given the test was positive.

P(X=+ | Y=-) = False Positive

P(X=- | Y=+) = False negative

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