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Solved: Consider the patient data in Example 2-8. Suppose
Chapter 3, Problem 139E(choose chapter or problem)
Consider the patient data in Example 2-8. Suppose that patients are randomly selected with replacement, from the total for hospital 4. Determine the following:
(a) Probability that the first patient admitted is the first one selected.
(b) Probability that four or fewer patients are selected to admit two.
(c) Expected number of patients selected to admit 10.
Questions & Answers
QUESTION:
Consider the patient data in Example 2-8. Suppose that patients are randomly selected with replacement, from the total for hospital 4. Determine the following:
(a) Probability that the first patient admitted is the first one selected.
(b) Probability that four or fewer patients are selected to admit two.
(c) Expected number of patients selected to admit 10.
ANSWER:
Answer
Step 1 of 3
(a)
Consider the patient data in given below table.
Suppose that patients are randomly selected with replacement, from the total for hospital
Hospital |
|||||
1 |
2 |
3 |
4 |
Total |
|
Total |
5292 |
6991 |
5640 |
4329 |
22,252 |
LWBS |
195 |
270 |
246 |
242 |
953 |
Admitted |
1277 |
1558 |
666 |
984 |
4485 |
Not Admitted |
3820 |
5163 |
4728 |
3103 |
16,814 |
We are asked to find the probability that the first patient admitted is the first one selected.
We need to find the probability that the patients are selected from the total for hospital
.
We can write from the table,
Let denote the number of patients selected with replacement until the first patient is reported as admitted.
We need to find
Hence follows a geometric distribution which we can write,
In a series of Bernoulli trials (independent trials with constant probability of a success), the random variable
that equals the number of trials until the first success is a geometric random variable with parameter
and
Hence the probability is,
Hence the probability that the first patient admitted is the first one selected is