1. Probability Review-1 Probability Review

2. Probability Review-2 Probability Theory Mathematical description of relationships or occurrences that cannot be predicted precisely An experiment is an activity whose outcome is subject to random (i.e. chance or unknown) variation. Examples: –Flip a coin –Toss a die –…

3. Probability Review-3 Sample Space The set of all possible outcomes of an experiment is known as its sample space, denoted as S Examples: –Coin flipping S = { } –Tossing a die S = {}

4. Probability Review-4 Events An event is a collection of outcomes from a sample space, denoted as E Examples: –Coin flipping Event = Get a tail –Tossing a die Event = Get an even number : outcome = {2,4,6} An event E is said to occur if one of the outcomes with which it is associated is realized during a replication of the experiment

5. Probability Review-5 Events Two events, E and F, are said to be mutually exclusive if E ∩ F =  which means… The complement of event E, denoted E c, is that unique set such that E U E c = S and E ∩ E c = 

6. Probability Review-6 Random Variable A random variable is a function that maps the sample space to the real line. Examples: –Coin flipping X =1 if heads 0 if tails –Tossing a die W = (the number that shows on die) A random variable is discrete if the possible values it can assume can be counted A random variable is continuous if it can assume any value in a continuous subset of the real line

7. Probability Review-7 Probability The probability associated with a particular event E, denoted P(E), can be thought of as representing the relative likelihood of that event occurring We will be generally thinking in terms of the probability of a random variable taking a specific value Examples: –Coin flipping P(X=1) = –Tossing a die P(W=6) = P(W=7) =

8. Probability Review-8 Axioms of Probability 0 ≤ P(E) ≤ 1 for any E P(S) = 1 If {E i, i=1,…,k} are mutually exclusive events, then

9. Probability Review-9 Probability Distributions Describes probabilities of values a random variable could take Discrete Examples: –Coin flipping P(X=x) =½ if x={0,1} 0 otherwise –Tossing a die P(W=w) = 1 / 6 if w={1,2,3,4,5,6} 0 otherwise Continuous Examples: –Altitude of an airplane Area under curve =

10. Probability Review-10 Common Probability Distributions Discrete –Discrete uniform –Poisson –Geometric –Binomial –… Continuous –Uniform –Exponential –Normal –Gamma –Beta –Triangular –…

11. Probability Review-11 PDF(PMF) vs. CDF Probability density function (p.d.f.) denote f Probability mass function (p.m.f.) denote f f(x)=P(X=x) Cumulative distribution function (c.d.f.) denote F F(x)=P(X≤x)

12. Probability Review-12 Mean and Variance Mean (Expected Value) E[X] =  = or Variance (Expected square distance from mean) Var(X) =  2 = E[(X-E[X]) 2 ] = E[X 2 ] – E[X] 2 Standard deviation (Spread)

13. Probability Review-13 Examples of Mean and Variance Coin flipping –Expected value = –Variance = –Standard deviation = Tossing a die –Expected value = –Variance = –Standard deviation = Continuous uniform between 0 and 2 –Expected value = –Variance = –Standard deviation =

14. Probability Review-14 Conditional Probabilities Consider two experiments with S 1 ={E 1,…,E m } and S 2 ={F 1,…,F n } P(E|F) = P{experiment 1 gets outcome E given that experiment 2 gets outcome F} Example: P(Ice cream sales > 10 cones | temperature = 85 F)

15. Probability Review-15 Example 1 of Conditional Probabilities The king comes from a family of 2 children. What is the probability that the other child is his sister?

16. Probability Review-16 Example 2 of Conditional Probabilities 52% of the students at a certain college are females. 5% of the students in this college are majoring in computer science. 2% of the students are women majoring in computer science. If a student is selected at random, find the conditional probability that 1)this student is female, given that the student is majoring in computer science; 2)this student is majoring in computer science, given that the student is female.

17. Probability Review-17 Bayes’ Theorem

18. Probability Review-18 Example 1 of Bayes’ Theorem Suppose that an insurance company classifies people into one of three classes – good risks, average risks, and bad risks. Their records indicate that the probabilities that good, average, and bad risk persons will be involved in an accident over a 1-year span are, respectively, 0.05, 0.15, and 0.30. If 20% of the population are “good risks”, 50% are “average risks”, and 30% are “bad risks”, what proportion of people have accidents in a fixed year? If policy holder A had no accidents in 1987, what is the probability that he or she is a good risk?

19. Probability Review-19 Example 2 of Bayes’ Theorem Suppose that there was a cancer diagnostic test that was 95% accurate both on those that do and those that do not have the disease. If 0.4% of the population have a cancer, compute the probability that a tested person has cancer, given that his or her test result indicates so.