Ch 2 - Probability Flashcards
Probability
The probability of an outcome is the proportion of times the outcome would occur if we observed the random process an infinite number of times.
Law of Large Numbers
As more observations are collected, the proportion Pn of occurrences with a particular outcome converges to the probability P of that outcome.
Disjoint/ Mutually Exclusive Events
Two outcomes that cannot both happen
Addition Rule of Disjoint Outcomes
Two-outcomes: P(A1 or A2) = P(A1) + P(A2),
k-outcomes, the probability that one of these will occur is P(A1) + P(A2) + … + P(Ak)
General Addition Rule
If A and B are any two events, disjoint or not, then the probability that at least one of them will occur is P(A or B) = P(A) + P(B) - P(A and B)
Probability Distribution
A table of all disjoint outcomes and their associated probabilities. Must satisfy (1) the outcomes are disjoint, (2) each probability is between 0 and 1, and (3) the probabilities must total 1
Multiplication Rule for Independent Processes
If A and B represent events from two different and independent processes, then the probability that both A and B occur is: P(A and B) = P(A) * P(B).
For k event A1, A2, …, Ak, the probability that they all occur = P(A1) * P(A2) * … * P(Ak)
Marginal and Joint Probabilities
If a probability is based on a single variable, it is a marginal probability. The probability of outcomes for two or more variables or processes is called a joint probability.
Conditional Probability: Outcome of Interest, Condition
The condition is the information we know to be true, ie “given” the condition is true or already happened.
Conditional Probability
The conditional probability of the outcome of interest A given condition B is: P(A | B) = P(A and B) / P(B)
General Multiplication Rule
If A and B are two outcomes or events, then, P(A and B) = P(A | B) * P(B). A = Outcome of Interest and B = Condition
Sum of Conditional Probabilites
Let A1, …., Ak represent all the disjoint outcomes for a variable or process. Then if B is an event, possibly for another variable or process, then:
P(A1 | B) + … + P(Ak | B) = 1
and
P(A | B) = 1 - P(A^c | B)
Bayes’ Theorem
P(outcome A1 of variable 1 | outcome B of variable 2) =
P(B|A1)P(A1) ÷
P(B|A1)P(A1) + … + P(B|Ak)P(Ak), where A1, A2, A3,…,Ak represent all possible outcomes of the first variable
Expected Value of Discrete RV
X takes outcomes x1, x2, …, xk with probabilities P(X=x1), …., P(X=xk), the expected value of X is the sum of each outcome multiplied by its probability:
E(X) = x1P(X=x1) + … + xkP(X=xk)
E(X) = ∑xi*P(X=xi)
