STAT 3401, Intro. Prob. Theory 11/21 Jaimie Kwon
1/6/2005
2.5 Calculating the probability of an event: the sample-point
method
• To find the probability of an event by the sample-point method, use the following steps:
o Define the experiment and clearly determine how to describe one simple event
o List the simple events associated with the experiment and test each to make certain
that it cannot be decomposed. (this defines the sample space S)
o Assign reasonable probabilities to the sample points in S, making certain that P(Ei)≥0
and ΣP(Ei)=1.
o Define the event of interest, A, as a specific collection of sample points. (A sample point
is in A if A occurs when the sample point occurs. Test all sample points in S to identify
those in A.)
o Find P(A) by summing the probabilities of the sample points in A.
• Example 2.2. An employer “randomly” selects two applicants for a job from five employer,
with competence 1 (best) to 5 (worst). Evaluate:
P(the employer selects the best and one of the two poorest applicants)=
P(the employer selects at least one of the two bests)=
• Example 2.3. In a balanced coin tossing, compute P(exactly 2 of 3 tosses are heads)
• Example 2.4. When A and B plays tennis, the odds are two to one that A wins. If A and B
plays two match, evaluate P(A wins at least one match).
• The method is direct and powerful (a bulldozer approach)
• Difficulties:
o Prone to human errors: forget some sample points in S
o Many sample spaces contain a very large number of sample points
• Detours:
o Many sample spaces generated by experimental data contain subset of sample points
that are equiprobable.
o Use a computer to list the large number of sample points and sum their probabilities
o The mathematical theory of counting, called “combinatorial analysis”
• HW: some questions from exercises 2.18~ 2.26
• Keywords: the sample-point method
