S1 DAPR1 lecture 7: probability rules and conditional

3 rules of probability

1. for any event the probability must fall between 0 and 1. 2. the sum of the probabilities of all the possible outcomes = 1. 3. probability of the complement of A (not A, A^c)

meaning of 0 ≤ P(A) ≥ 1

for any event the probability must fall between 0 and 1

meaning of: P(A1) +... P(Ai) = 1

the sum of all the probabilities of all possible outcomes = 1

meaning of : **P(~A) = 1 (minus) P(A)**

probability of the complement of A (not A, A^c)

for all events: P(A union B) =

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

for mutually exclusive events P(A n B) =

0

therefore fir mutually exclusive events P (A union B) =

P(A) + P(B)

if A and B occur independently (one doesn’t affect the other), then the probability of A and B occurring together can be found using the following equation:

P(A n B) = P(A)P(B)

multiplication rule for conditional probabilities

P(A n B) = P(A)P(B|A) = P(B n A) = P(B)P(A|B)

P(A|B) meaning

probability of B given A (has occurred)

division rule for conditional probabilities

P(B|A) = P(A n B)/P(A); Or the inverse: P(A|B) = P(A n B)/P(B)

Bayes rule

a formula for conditional probability when P(A n B) is not known

how is Bayes formula derived?

same as division equation but replace P(A n B) with P(B)P(A|B)

Bayes formula:

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

how to asses if events are independent

use the multiplication rule and contingency tables. tabulate the proportions and think about probabilities, assume all events are independent then calculate the expected frequencies (products of the 2 probabilities)

therefore, if variables are independent

the probabilities given that something for each should be the same

if variables are not independent

the given that probabilities will be different - determining that the event in the conditional probability has an impact on the probability, meaning that it is not independent.

how is it useful for statistical test

this forms the fundamental background principles of the chi-squared test of independence - allows us to determine if 2 categorical variables of nominal data or dependent or not