STAT1127-Some Basic Probability Concepts

–an event can occur in N mutually exclusive and equally likely ways

–and M of these ways possess a trait, E

classical probability

–Some process is repeated a large number of times, n

–and some resulting event with the characteristic E occurs m times

relative frequency probability

venn diagram

the union of 2 sets

the intersection of 2 sets

complementary set

Given some process (or experiment) with n mutually exclusive outcomes (events), E₁, E₂, …, E_n, the probability of any event E_i, i=1, 2, …, n, is assigned a nonnegative number.

elementary properties of probability I

•The sum of the probabilities of the mutually exclusive outcomes is equal to 1.

elementary properties of probability II

•Consider any two mutually exclusive events, E_i and E_j. The probability of the occurrence of either E_i or E_j is equal to the sum of their individual probabilities.

elementary properties of probability of III

–The probability result based on the whole sample

Marginal Probability

–The probability result based on a subset of the sample

conditional probability

•The probability that a randomly selected subject possesses two or more characteristics at the same time

joint probability

•The conditional probability of A given B is equal to the probability of A∩B divided by the probability of B, provided the probability of B is not zero

conditional probability

venn diagram

independent events

P(B|A)=P(B), P(A|B)=P(A)

Given some variable that can be broken down into m categories: A₁, A₂, …, A_m, and another jointly occurring variable that is broken down into n categories: B₁, B₂, …, B_n, the marginal probability of A_i, P(A_i), is equal to the sum of the joint probability of  with all the categories of B.

marginal probability

results when a test indicates a positive status when the true status is negative

false positive

results when a test indicates a negative status when the true status is positive

false negative

the probability of a positive test result given the presence of the disease

sensitivity, P(T|D)

•the probability of a negative test result given the absence of the disease

specificity, P(T-|D-)

•the probability that a subject has the disease given that the subject has a positive screening test result

positive predictive value P(D|T)

the probability that a subject does not have the disease, given that the subject has a negative screening test result

negative predictive value P(D-|T-)

bayes theorem

P(A|B)=P(B|A)P(A)/P(B|A)P(A)+P(B|A-)P(A-)

P(A|B)

P(A n B)/P(B)

P(B)

P(A n B)+P(A- n B)

P(A n B)

P(B|A)P(A)

P(A- n B)

P(B|A-)P(A-)