TOPIC #6: Understanding chance

Module 3 DATA1001

What is the prosecutor’s fallacy?

a mistake in statistical thinking, where it is assumed that the probability of a random match is equal to the probability that the defendant is innocent.

What is chance (frequentist defintion)

the percentage of time a certain event is expected to happen if the same process is repeated long term.

what are the basic properties of chance?

1. P(impossible event) = 0, P (certain event) = 1

2. P(event) = 1 - P(Complement event)

3. Drawing at random means that a collection of objects have equal chances being picked.

Conditional probaility

chance that a certain event occurs given, another event has occurred

Multiplication principle

The probability that 2 events occur is the chance of the 1st event multiplied by the chance of 2nd event, given 1st has occurred.

Independence

2 events are independent if the chance of the 2nd given the 1st is the same as the 2nd.

How do you ensure independence

drawing randomly with replacement ensures independence

how does the multiplication principle apply to cases were two events are independent

the chance of both occurring is the product of their unconditional probabilities.

dependence

two events are dependent if the chance of the 2nd given the 1st is not the same as the chance of the 2nd as it depends on the result of the 1st event

how do you ensure dependence?

drawing without replacement

mutually exclusive

two things are mutually exclusive when the occurrence of one event prevents the other

event 1 occurs p(event 2) = 0

difference between mutually exclusive and independence

if something is mutually exclusive, then the chance of event 1 happening, prevents event 2, however, independence does not chance the chance of event 2 .

addition rule

if two things are mutually exclusive, then the chance of at least 1 occurring is the sum of the individual chances
P (at least 1 of 2 events occurs) = p(Event 1) + p(event 2)

Binomial coefficients

the number of ways of rearranging the n objects is given by the binomial coefficient —>

what is a binary trial?

where only two things can occur so P(event)= p p(not event) = 1 - p