Читать книгу Applied Biostatistics for the Health Sciences - Richard J. Rossi - Страница 63

Determining the probabilities associated with complex real-life events often requires a great deal of information and an extensive scientific understanding of the structure of the chance experiment being studied. In fact, even when the sample space and event are easily identified, the determination of the probability of an event can be an extremely difficult task. For example, in studying the side effects of a drug, the possible side effects can generally be anticipated and the sample space will be known. However, because humans react differently to drugs, the probabilities of the occurrence of the side effects are generally unknown. The probabilities of the side effects are often estimated in clinical trials.

The following basic probability rules are often useful in determining the probability of an event.

1 When the outcomes of a random experiment are equally likely to occur, the probability of an event A is the number of outcomes in A divided by the number of simple events in S. That is,

2 For every event A, the probability of A is the sum of the probabilities of the outcomes comprising A. That is, when an event A is comprised of the outcomes O1,O2,…,Ok, the probability of the event A is

3 For any two events A and B, the probability that either event A or event B occurs is

4 The probability that the event A does not occur is 1 minus the probability that the event A does occur. That is,

Example 2.20

Table 2.8 gives a breakdown of the pool of 242 volunteers for a university study on rapid eye movement (REM). Use Table 2.8 to determine the probability that

Table 2.8 Summary Table for the n=242 Volunteers in a University Study of Rapid Eye Movement (REM)

Gender	Age
	<18	18–20	21–25	>25
Female	3	58	42	25
Male	1	61	43	9

1 a female volunteer is selected,

2 a male volunteer younger than 21 is selected,

3 a male volunteer or a volunteer older than 25 is selected.

Solutions Let F be the event a female volunteer is selected, M the event a male volunteer is selected, 21 the event a volunteer younger than 21 is selected, and 25 the event that a volunteer older than 25 is selected. Since a volunteer will be selected at random, each volunteer is equally likely to be selected, and thus,

1 the probability that a female volunteer is selected is

2 the probability that a male volunteer younger than 21 is selected is

3 the probability that a male volunteer or a volunteer older than 25 is selected is

Example 2.21

Use table of percentages for blood type and Rh factor given in Table 2.9 to determine the probability that a randomly selected individual

Table 2.9 The Percentages of Each Blood Type and Rh Factor

Blood Type	Rh Factor
	+	−
O	38%	7%
A	34%	6%
B	9%	2%
AB	3%	1%

1 has Rh positive blood,

2 has type A or type B blood,

3 has type A blood or Rh positive blood.

Solutions Based on the percentages given in Table 2.9

1 Rh positive blood is O+, A+, B+, and AB+, and thus, the probability that a randomly selected individual has Rh positive blood is

2 type A or type B blood is A+, A−, B+, B−. Thus,

3 the probability an individual has type A blood or Rh positive blood is