Probability Flashcards
What is probability
- looking at the past
- looking at average of occurrences of past events
-assess probability of future events based on how often it happened in the past
types of probability
Theoretical -> can be determined accurately based on logic on e.g., odds of getting a spade in a deck of cards (1/4)
empirical -> relies on past observation eg., probability of a meteor hitting earth
objective -> in between
eg., probability of conceiving twins
-relies on how often it happens
-still relying on past data but happens reliably enough that its considered accurate
subjective -> cant be determined accurately
-happens rarely, not enough data
probability experiment, uncertain outcome, trial
taking observations in an environment with an uncertain outcome
-outcome of the experiment that have a degree of chance eg., coin toss
trial - repetition of an experiment
what types of outcomes are there in a probability experiment
outcome or sample space
-set of possible outcomes
outcomes
- can be mutually exclusive, can only have one outcome at a time
-exhaustive, includes all possible outcomes
-if its both, its a simple outcome
random vs fixed variable
random variable
-characteristic that were measuring with the uncertain outcome eg., what side the coin lands on
-random based on the context in which it is measured (each coin is slightly variable (diameter, etc)
-these arent considered random variables, the coin toss is the variable, not the coin itself. the coins are assumed to be constant
fixed variable
-non random variable in the experiment
-the coin in the coin toss
defining a sample space
coin toss, variable of interest is the side of the coin facing up
-2 mutually exclusive outcomes
-its exhaustive since there are only 2 choices
-simple outcome
2 flips of a coin
-variable is the sequence of coin landing
-4 outcomes
-HH, HT, TH, TT
-simple outcomes
-use a tree diagram
->graph to ensure all simple outcomes are identified
rules of assigning probability to an outcome
probability of a simple outcome: 0 -> 1
sum of probabilities of simple outcomes = 1
simple outcomes of a sample space and their probabilities make up a probability distribution or probability distribution function (pdf)
-explains how total probability (1) is distributed amongst probable outcomes
continuous datasets
if data are quantitative and continuous, then determine relative frequency in interval of values
(how many heights between 54 and 58, 58 and 64)
composite outcomes
eg., rolling odd values in a dice roll
probability is the sum of the probabilities of the simple outcomes
P(1,3, or 5) =P(1) + P(3) + P(5) = 1/6 + 1/6 + 1/6 = 1/2
conditional probability
probability that one composite outcome will occur given that another composite outcome occurs
p of A given B
P(A|B) = no. simple outcomes in both A and B / no. of simple outcomes in B
P(A|B) = P(A and B) / P(B)
multiplication rule
probability that outcomes a and b will occur, rearrange conditional probability eqn
P(A and B) = P(A|B)P(B)
addition rule
probability that A or B will occur
P(A or B) = P(A) + P(B) - P(A and B)
What is independence in probability
if the probability of outcome A is the same whether or not B occurs, A and B are independent
P(A|B)=P(A)
P(AB) = P(A)P(B)
Central Limit Theorem
If x has a distribution with a mean = μ & standard deviation = σ with
or without a normal shape, then the sampling distribution of the
mean, x̄ based on random samples of size n:
* Tends to be normal as sample size increases
* Mean = μ
Expected Value
Weighted mean by probability
sampling distribution
how far is the sampling mean from the population mean
if x is normally distributed with a mean=µ, then the sampling deviation of the mean x̄ based on a sample size n, then:
Mean = μ
standard deviation:
σ⌄x̄ = σ/√n
