•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

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

0)(

1)(0

P

AP

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

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

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

!!

kn

nn

kP

!!

!

knk

n

k

nC n

k

Sampling with replacement?

yes no

Order is important?

yes S2={RB, BR, GB, BG, GR, RG, RR, BB, GG}

S3={RB, BR, GB, BG, GR, RG}

no S4={RB, BG, GR, RR, BB, GG}

m=6

S4={RB, BG, GR}

932 knm

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 (S1={R, B, G}, n=3)

IME 312

332 Cm

632 Pm

Word Problem Translation in ProbabilityProb. 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