Random Experiment

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
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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(AB) P(A) Then
P(A B) P(A) P(B)
IME 312
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• Conditional Probability
• P(A B) P(A B) / P(B)
• Bayes Theorem
• P(AB) P(BA) P(A) / P(B)
• (Refer to the example in your Supplementary
  Notes)

IME 312
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• 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
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Counting number of elements in a Sample Space
(m) Assume selecting k items from a Space with n
items. For example Pick 2 pens (k2) from a box
that has one pen of each colors Red, Black, and
Green (S1R, B, G, n3)
Sampling with replacement? Sampling with replacement?
yes no
Order is important? yes S2RB, BR, GB, BG, GR, RG, RR, BB, GG S3RB, BR, GB, BG, GR, RG
Order is important? no S4RB, BG, GR, RR, BB, GG m6 S4RB, BG, GR
IME 312
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Word Problem Translation in
Probability Prob. exactly a P(X
a) Prob. less than a P(X lt a) Prob.
more than a P(X gt 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 a1) .. P(X b)
IME 312