1. Random Experiment Random Variable: Continuous, Discrete Sample Space: S Event: A, B, E Null Event Complement of an Event A’ Union of Events (either, or) A B Intersection of Events (both, and) A B Mutually Exclusive Events A B = Independent Events Venn diagram IME 312

2. Probability Probability is the likelihoods (frequency) of an event to happen. Probability is the language and mathematical foundation of inferential statistics. P(....) Probability Properties P(S) = 1 IME 312

3. Additive Rule P(A B) = P(A) + P(B) - P(A B) If A and B are mutually exclusive, then P(A B) = P(A) + P(B) Total Probability Rule P(B) = P(B A) + P(B A’ ) Probability of Independent Events If P(A|B) = P(A) Then P(A B) = P(A) * P(B) IME 312

4. Conditional Probability P(A | B) = P(A B) / P(B) Bayes’ Theorem P(A|B) = P(B|A) * P(A) / P(B) (Refer to the example in your Supplementary Notes) IME 312

5. Number of elements in a sample space can be calculated using the following formulas for each setting of experiments mentioned below. Permutation Experiment: Select k objects from n objects, without replacement, order is important, i.e. ab and ba Combination Experiment: Select k objects from n objects, without replacement, order is not important, i.e. ab or ba and not both IME 312

6. Sampling with replacement? yesno Order is important ? yes S 2 ={RB, BR, GB, BG, GR, RG, RR, BB, GG} S 3 ={RB, BR, GB, BG, GR, RG} no S 4 ={RB, BG, GR, RR, BB, GG} m=6 S 4 ={RB, BG, GR} Counting number of elements in a Sample Space (m) Assume selecting k items from a Space with n items. For example: Pick 2 pens (k=2) from a box that has one pen of each colors Red, Black, and Green (S 1 ={R, B, G}, n=3) IME 312

7. Word Problem Translation in Probability Prob. exactly a P(X = a) Prob. less than a P(X < a) Prob. more than a P(X > a) Prob. at most a P(X  a) Prob. at least a P(X  a) = 1 - P(X  a) Prob. between a & b P(a  X  b) = = P(X  b) - P(X  a) = or = P(X = a) + P(X = a+1) +.. + P(X = b) IME 312