Chapter 3.1-3.2 Probability Flashcards
Random phenomenon
Situation in which the outcome is uncertain but there would be a definite distribution of ootsong if the situation was repeated many times under identical outcomes
Law of large Numbers
Random behavior isn’t typically predictable in short term, usually very predictable in the long term. “
Probability
Prob of an outcome is the proportion of times the outcome would occur if we observed the random process a infinite sober of times
True probabilities usually given is a table with some rules.
The prob of an event is the Sum of the probabilities for outcomes that makeup the event
Sample space
Set of all possible outcomes (omega or Ω )
Events
Subsets of outcomes _ all outcomes one events, not all events are outcomes
Probability distribution
A table of all disjoint outcomes and their associated probabilities
Outcomes must be disjoint, prob. Must be between 0-1 and total 1.,
- Describe a phenomenon.
- Describe all possible outcomes (the sample space Ω)
- Assign a probability to each possible outcome.
Disjoint
Disjoint events are mutually exclusive, both can’t happen
These are dependent
Addition rule of disjoint outcomes
P(A1 or A2) = P (A1) + p (A2)
General addition rule
P(a&b) = p(a) + P(B) - p (a &b)
IF A & B are any two events, disjoint or not, this is the formula for the prob of at least one of then occurring.
Where p(a & b) is the probability of them both occurring.
Complement
The complement of on event is all other events not in that event’.
To calculate, 1-P(event complement)
Independent events and the multiplication rule
When knowing one outcome provides no useful information about the other.
When A & B are events from independent processes, the probability that both world occur is
P (a&b) = P(a) * p(b)
Probability rules
I. For any event A, <= P(A) <=1
2. P(Ω) = 1
3. If events A & b are disjoint then p (a or b) = p(a) + p(b)
3. B, if A & B are not disjoint then p (a or b) = p(a) + p(b) - p(a and b)
4. If Ac denotes the complement of A, then P(AC) = 1 - p(a)
5. If A & b are independent then p(a&b) =p(a) * p(b)
Marginal & joint probabilities
Marginal — Based on a single variables, focused on outer total data in a contingency table
Joint – based on two variables,
Conditional probability
The conditional probability of outcome A given outcome B is
P(A|B) = p(a and b)/p(b)
In a formula | indicates given
The variable on the left is what is being computed given the variable on the right
If events are independent then P( A | B) = P (A)
General multiplication rule for conditional probabilities
If A & B represent two outcomes or events, then
P(A and B) = p(a|b) * p(b) = (p(a and b)/p(b)) * p(b)
A- outcome of interest
B- condition
Rules for conditional probabilities
If A1 and A2 an disjoint events, if B is another event then
P(a1 or a2 | b) = p(a1|b) + p(a2|b)
If A is an event then
P( Ac|b) = 1 - p(A|B)
Tree diagrams
tools to organize outcomes & probabilities around the structure of the data - useful when two or more processes occur in a sequence & each process is conditioned on its predecessors
Bayes‘ theorem
A mathematical rule for inverting conditional probabilities, allowing one to find the probability of a cause given its effect
P(outcome A1 of variable 1 | outcome B of variable 2), Bayes’ theorem states that this conditional prob. Can be identified as the following fraction
P(B|A1)* P(A1)
———————————————
P(B|A1)* P(A1) + P(B|A2)* P(A2) + …. P(B|Ak)* P(Ak)
Where A2, A3,…Ak represent all other possible outcomes of the first variable
