Please enable JavaScript.
Coggle requires JavaScript to display documents.
Chapter 6 - Probability theory - Coggle Diagram
Chapter 6 - Probability theory
Assigning probability to events
Mutually exclusive => No two outcomes can occur at the same time
Event => A set of outcomes
Collectively exhaustive => All possible outcomes must be included
Probability => The chance that an event will occur
List the outcomes, the list must be collectively exhaustive and mutually exclusive
Sample space => The set of all possible outcomes of a random experiment
An experiment of which the outcome is not known in advance
Requirements probabilities must meet:
The probability of any outcome must lie between 0 and 1
1
The sum of the probabilities of all outcomes must be 1
2
Random experiment:
How do we obtain P(Oi)?
3 approaches to assessing the probability of an uncertain event:
a priori classical approach
empirical classical approach
Subjective probability
Probability of a joint event, A and B
8
Simple event
An outcome from a sample space with one characteristic
Multiplication rules
P(AnB) = P(A|B) x P(B)
If A and B are independent then P(AnB) = P(A) x P(B)
Priori classical approach
3
Empirical classical approach
4
Mutually exclusive
7
If P(AnB) = 0 then events A and B are mutually exclusive
P(AnB) => A and B
Joint event
Involves two or more characteristics simultaneously
Complement of an event A (denoted A')
All outcomes that are not part of event A
Subjective approach
An individual judgement or opinion about the probability of occurrence is made
The probability will differ from one person to the next
Marginal (simple) probability
P(A) = P(AnB1) + P(AnB2) + ...........+ P(AnBk) where B1, B2,......, Bk are k mutually exclusive and collectively exhaustive events
9
Conditional probability
10
the conditional probabilities of B given that A has occurred
the conditional probabilities of A given that B has occurred
11
Bayes' theorem
12
13
Statistical independence
Two events are independent if and only if:
P(A|B) = P(A)
Events A and B are independent when the probability of one event is not affected by the other event
The law of total probability
P(AnB) = P(A|B1) x P(B1) + ........P(A|Bk) x P(Bk) where B1, B2 and Bk are k mutually exclusive and collectively exhaustive events
Counting rules
If k1 = k2 = .... = kn then the possible number of outcomes is equal to k^n
The number of ways that n items can be arranged in order is
n! = n(n-1)....(1)
If any one of Ki different mutually exclusive and collectively exhaustive events can occur on the ith of n trials, the number of possible outcomes is equal to k1 x k2 x ... kn
Permutations
The number of ways of arranging X objects selected from n objects in order is
14
Rules for counting the number of possible outcomes
Combinations
The number of ways of selecting X objects from n objects irrespective of order is
15
General addition rule
P(AuB) = P(A) + P(B) - P(AnB)
If A and B are mutually exclusive then P(AuB) = P(A) + P(B)
Visualizing collectively exhaustive events
Venn diagrams
5
Contingency tables
Tree diagrams
Visualizing collectively exhaustive events
The entire sample space is exhausted
6