Introduction to probability

Transcription

1 Introduction to probability P(A) = Probability of an event, A, occuring Before we start discussion distributions, let s take a step back and talk about some basic rules of probability. Probability is fundamentally about assigning probabilities to events. An event can be pretty much anything for which there is an alternative outcome. Eg) A = { sun rises tomorrow, Supreme Court nominee will be blocked, etc. }

2 Rules of probability P(A C ) = 1 P(A) P(A C ) Probability of something not happening. P(A) Probability of something happening.

3 Complements of events If P(Terrorism) = What is P(Terrorism c ) =?

4 Union of events P(A or B) = P(A) + P(B) A B Probability that one event or another event that are independent from each other is just their sum.

5 Union of events If P(Terrorism) = 0.01 and; P(Falcons win superbowl) = 0.05 What is: P(Terrorism or Falcons win superbowl) =?

6 Union of events If P(Terrorism) = 0.01 and; P(Falcons win superbowl) = 0.05 P(Terrorism or Falcons win superbowl) = P(Terrorism) + P(Falcons win superbowl) = = 0.06

7 Intersection of events P(A and B) = P(A)P(B) A B Probability of both independent events occurring is just their probabilities multiplied together.

8 Intersection of events If P(Terrorism) = 0.01 and; P(Falcons win superbowl) = 0.05 What is: P(Terrorism and Falcons win superbowl) =?

9 Intersection of events If P(Terrorism) = 0.01 and; P(Falcons win superbowl) = 0.05 P(Terrorism and Falcons win superbowl) = P(Terrorism)P(Falcons win superbowl) = (0.01)(0.05) =

10 Probability distributions for discrete and continuous variables Probability distributions are full distributions of all possible outcomes and probability of those outcomes occurring. Recall that discrete variables and variables which take on a finite number of values. Continuous variables take on a theoretically infinite number of values.

11 Probability distributions for discrete and continuous variables Discrete probability distributions are probability distributions which assign a probability to each individual outcome. Continuous probability distributions are probability distributions which assign probabilities to intervals.

12 Probability distribution of a discrete variable Children in families = y ={4, 6, 2, 1, 1, 2} The number of children in families is a good example of a discrete variable.

13 Probability distribution of a discrete variable Children in families = y ={4, 6, 2, 1, 1, 2} 0 P(y) 1 N P(y) = 1 i=1 P(y = 4) = 1/6, P(y = 6) = 1/6, P(y = 2) = 2/6,P(y = 1) = 2/6 This is the full probability distribution of y.

14 Probability distribution for continuous variables Continuous variables have a theoretically infinite continuum of values. Strangely enough, because of this continuous distributions always assign probabilities to ranges rather than values.

15 Probability distribution for continuous variables: IQ scores Let s take IQ scores again as an example. Each IQ range corresponds to a probability value.

16 Probability distribution for continuous variables: IQ scores P(100 > IQ > 160) = 0.50 P(100 < IQ < 40) = 0.50

17 Probability distribution for continuous variables: IQ scores These probability values actually directly correspond to the rule if the data follow a normal distribution.

18 Probability distribution for continuous variables: IQ scores x = 100, s = 15 P(85 > IQ > 115) = 0.68

19 Probability distribution for continuous variables: IQ scores x = 100, s = 15 P(70 > IQ > 140) =?

20 Probability distribution for continuous variables: IQ scores x = 100, s = 15 P(70 > IQ > 140) = 0.95

21 Probability distribution for continuous variables: IQ scores x = 100, s = 15 P(55 > IQ > 155) =?

22 Summarizing probability distributions At their core, probability distributions are just functions just like those you might remember from calculus: ie f (x) = x 2 They are functions that are defined, however, by their parameters. These parameters are typically the mean and variance

23 Summarizing probability distributions Normal distribution= f (µ, σ) = N(µ, σ) Normal distribution= f ( x, s) = N( x, s) For example, where the normal distribution lies on the x axis depends upon it s mean, or expected value. How fat the normal distribution is depends on its standard deviation (or variance which is just s 2

24 Summarizing probability distributions V Clinton,GAcounties N(36, 5) When we describe a variable in terms of its distribution, we usually specify what kind of distribution it follows, the mean and the standard deviation of that distribution. V Clinton,GAcounties variable is the county vote share for Hillary Clinton in the 2016 election for the 159 counties in GA.

25 Clinton vote share in GA counties

26 Clinton vote share in GA counties

27 Trump vote share in GA counties

28 Trump vote share in GA counties

29 Trump vote share in GA counties Def: If V Trump,GA is a variable containing Trump vote share for GA counties. Def: And V Trump,GA N(60, 5) Q: What are a and b in the equation P(a < V Trump,GA < b) = 0.95? What does this mean, in words?

30 Trump vote share in GA counties Def: If V Trump,GA is a variable containing Trump vote share for GA counties. Def: And V Trump,GA N(60, 5) Q: What are a and b in the equation P(a < V Trump,GA < b) = 0.95? What does this mean, in words? A: a = = 50, b = = 70.

31 Trump vote share in GA counties Def: If V Trump,GA is a variable containing Trump vote share for GA counties. Def: And V Trump,GA N(60, 5) Q: What is p in the equation P(55 < V Trump,GA < 65) = p? What does this mean, in words?

32 Trump vote share in GA counties Def: If V Trump,GA is a variable containing Trump vote share for GA counties. Def: And V Trump,GA N(60, 5) Q: What is p in the equation P(55 < V Trump,GA < 65) = p? What does this mean, in words? A: p = 0.68.