Probability

Probability Probability
Part of everyday life  probability in poker in lottery  probability of rain  probability Guides our expectations Lies at the heart of statistical inference
Whenever we generalize from a sample to a population, we do so with a probability statement

2 Definition of Probability
Theoretical Definition - Analytic View used if we can list all possible outcomes of an event and if each outcome is equally likely.
The p (a ) = probability of event A is: N um ber of outcom es favorable to A T otal N um ber of Possible O utcom es
Example: What is the probability of rolling a 4 with 2 6-sided dice? 3 1 Die 1 Possible Outcomes for 2 Fair Dice 6 5 4 3 2 1 7 6 5 4 3 2 1 8 7 6 5 4 3 2 9 10 11 12 8 9 10 11 7 8 9 10 6 7 8 9 5 6 7 8 4 5 6 7 3 4 5 6 Die 2 There are 3 ways to roll a 4 with 2 dice: p(4) = 3/36 = 1/12 4 Definition of Probability
Empirical definition of probability Relative Frequency View  Used when you don’t know all the possible outcomes or they are not equally likely.  In this case, we define probability empirically, by observation: p (a ) = N um ber of tim es A occurs Large num ber of trials A much more mechanical approach to probability  Roll dice 100 times and count how many 4s 5 More Definitions
Event – the outcome of a trial or probabilistic experiment
Mutually Exclusive - events that cannot happen at the same time  e.g.  P(A  e.g.
Getting a King in a deck of 52 cards and B) = 0 P(Male and Female) = 0 Exhaustive – A set of events that represents all possible outcomes  e.g. P(Male or Female) = 1 6 2 More Definitions
Independent Events – one event does not have an impact on the probability of the occurrence of another
Sampling with replacement – the result of any event is replaced before the next event  Draw a card out of a deck of 52 and put it back before drawing again  e.g P(King of Hearts) = 1/52 for every draw 7 One more definition
Sampling without replacement - the result of any event is not replaced before the next event  Draw a card out of a deck of 52 and leave it out before drawing again  e.g. P(King of Hearts) = 1/52 for first draw and P(King of Hearts) = 1/51 for second, etc.  e.g. Raffle Drawings 8 Probabilities and Proportions By either definition (theoretical or empirical), probabilities are proportions
In inference, probabilities are often treated as expected proportions
 We would expect half the number of coin flips to be heads  We expect 51% of marriages to end in divorce 9 3 Probabilities and Proportions Probabilities range between zero and one (like any proportion)
Probability of an impossible event is 0
 P(3
inches of snow in LA) = 0 Probability of a “sure thing” is 1.  P(Smog in LA) = 1 10 Probabilities and Proportions
Probability of an event not happening = 1 – the probability of the event happening  P(not Ace) = 1 - P(Ace) Ace) = 1 – 4/52 or 1/13  P(not Ace) = 1 – 1/13 = 12/13  P(not 11 Laws of Probability
Additive Law of Probability –  Given mutually exclusive events: P(A or B) = P(A) + P(B)  See OR, think ADD
Multiplicative Law of Probability –  Given independent events P(A and B) = P(A) x P(B)  See AND, think MULTIPLY  This is called the joint probability  If the events are dependent this gets more complicated 12 4 Examples
In a normal deck of 52 cards what is the probability of :  Drawing an ace?  Drawing a spade?  Not drawing a spade?  Drawing the ace of spades?  Drawing an ace and a spade?  Drawing an ace or a spade?  Drawing 2 aces in a row?  Drawing 2 spades in a row?  Drawing a spade given you’ve drawn a black card? 13 Conditional Probability
This if the probability that an event will occur, given that some other event has occurred.  P(A | B) = P(Spade | Black Suit) = ??? Knowing you selected a black suit, what is the probability of a spade?  The conditional property usually works to reduce the total number of events  P(Red Suit | Ace) = ??  P(Ace | King) = ?? 14  i.e. x Probability Distributions for Discrete Variables p Probability Distribution for the Possible Events when Rolling 2 Dice 12 1/36 11 2/36 10 3/36 0.18 9 4/36 8 5/36 0.16 0.14 7 6/36 6 5/36 5 4/36 4 3/36 3 2/36 2 1/36 p(x) 0.12 0.1 0.08 0.06 0.04 0.02 0 2 3 4 5 6 7 8 9 10 11 12 Value of x 15 5 Probability Distributions for Discrete Variables x p 12 1/36 11 2/36 10 3/36 9 4/36 8 5/36 7 6/36 6 5/36 5 4/36 4 3/36 3 2/36 2 1/36 Thus, the probability that x falls in the interval between any two numbers a and b, inclusive, is found by simply summing the probabilities for x over all possible values between a and b, inclusive. p(a ≤ x ≤ b) = sum of p(x = c) for all values c such that: a ≤ c ≤ b. For example, the probability that x is between 3 and 5, inclusive is: p(3 ≤ x ≤5) = p(3) + p(4) +p(5)= 2/36 +3/36 +4/36 = ¼ =.25. 16 Probability Distributions for Discrete Variables x p 12 1/36 11 2/36 10 3/36 9 4/36 8 5/36 7 6/36 6 5/36 5 4/36 4 3/36 3 2/36 2 1/36 By the same argument, P(x ≤ a) = sum of p(x =c) for all c such that c ≤ a. For example, p(x ≤ 4) = = p(x = 4) + p(x=3) + p(x=2) = 3/36 + 2/36 + 1/36 = 1/6 = .1667. 17 Probability Density Functions for Continuous Variables
This was covered in the last section but to review: f (X ) =

2 1 (e ) − ( X − µ ) σ 2π 2σ 2 Relies on Mean and SD Allows us to calculate probability for a range of scores 18 6 Density over a range = Area = Probability Mean of 30, SD 10 Probability of scores falling below 27
Probability = .2743 (from Z table)

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