MAT 119 FALL 2001
MAT 119
FINITE MATHEMATICS
NOTES
PART 2 – PROBABILITY
CHAPTER 8
ADDITIONAL PROBABILITY TOPICS
8.1 Bayes’ Formula
Partition
A sample space S is partitioned into n subsets A1, A2, …, An, provided:
a. The intersection of any two of the subsets is empty.
b. Each subset is nonempty.
c. A1 È A2 È … È An = S
Bayes’ Formula
Let S be a sample space partitioned into n events, A1, … , An. Let E be any event of S for which P(E) > 0. The probability of the event Aj (j = 1, 2, … , n), given the event E, is
since
P(E) = P(E Ç A1) + P(E Ç A2) + … + P(E Ç An)
= P(A1).P(E\A1) + P(A2).P(E/A2) + … + P(An).P(E\An)
A priori probability – P(Aj) – before the fact
A posteriori probability – P(Ai/E) – after the fact
8.2 The Binomial Probability Model
Bernoulli Trial
Random experiments are called Bernoulli trials if
a. The same experiment is repeated several times.
b. The are only two possible outcomes, success and failure, on each trial.
c. The repeated trials are independent.
d. The probability of each outcome remains the same for each trial.
Binomial probability – n Bernoulli trials
Binomial Probabilities
In a Bernoulli trial the probability of exactly k successes in n trials is
where p is the probability of success and q = 1 – p .
8.3 Expected Value
Expected Value
Let S be a sample space and let A1, A2, …, An be n events of S that form a partition of S. Let p1, p2, …, pn be the probabilities of the events A1, A2, …, An, respectively. If each event A1, A2, …, An is assigned the payoff m1, m2, …, mn, respectively, the expected value E corresponding to these payoffs is
E = m1.p1 + m2.p2 + … + mn.pn
Steps for computing Expected Value
1. Partition S into n events A1, A2, …, An.
2. Determine the probability p1, p2, …, pn of each event A1, A2, …, An. Since these events constitute a partition of S, p1 + p2 + …+ pn = 1.
3. Assign payoffs m1, m2, …, mn to each event A1, A2, …, An.
4. Calculate E = m1.p1 + m2.p2 + … + mn.pn.
Expected Value for Bernoulli Trials
In a Bernoulli process with n trials the expected number of successes is E = n.p where p is the probability of success on any single trial.
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DOUGLAS A. WILLIAMS, ARIZONA STATE UNIVERSITY, DEPARTMENT OF MATHEMATICS