Download Elements of Statistics 1 Partition Formula

AMS 102.7 Spring 2006
Jingyu Zou
Elements of Statistics
Lecture Notes # 15
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Partition Formula
Definition 1.1 A set of events {Ei }ni=1 are called a partition of the sample space Ω if
(1) {Ei } are mutually disjoint;
(2) the union of {Ei } forms the sample space, i.e., ∪ni=1 Ei = Ω.
Example 1.2 For any event A, A and AC form a partition.
Theorem 1.3 Let {Ei }ni=1 be a partition of the sample space, then for any event F ,
P (F ) =
n
X
P (F |Ei )P (Ei )
i=1
Example 1.4 For a statistical test, we know that the probability H0 is true is p, and we know the
probability of two types of errors α, β. What is the probability to get an error in this test?
Example 1.5 In answering a question on a multiple-choice test, a student either knows the answer
or guesses. Let p be the probability that the student knows the answer and 1 − p the probability
that the student guesses. Assume that a student who guesses at the answer will be correct with
probability 1/m where m is the number of multiple-choice alternatives. What is the probability
that he answers correctly on this question?
Example 1.6 A certain disease is only associated with a pair of genes H, T . One may have gene
H and T independently with probability p and 1 − p. If one has genotype HH, he has the disease
with probability α1 . If one has genotype HT or T H, he has the disease with probability α2 . If
one has genotype T T , he has the disease with probability α3 . What is the probability that one has
that disease?
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