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# Classical, Relative Frequency, and Subjective Probability Definitions for Beginning Statistics

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By — McGraw-Hill Professional
Updated on Aug 12, 2011

Practice problems for these concepts can be found at:

The classical definition of probability is appropriate when all outcomes of an experiment are equally likely. For an experiment consisting of n outcomes, the classical definition of probability assigns probability to each outcome or simple event. For an event E consisting of k outcomes, the probability of event E is given by formula (4.4):

P(E) =                                     (4.4)

EXAMPLE 4.8   The experiment of selecting one card randomly from a standard deck of cards has 52 equally likely outcomes. The event A1 = {club} has probability , since A1 consists of 13 outcomes. The event A2 = {red card} has probability , since A2 consists of 26 outcomes. The event A3 = {face card (Jack, Queen, King)} has probability , since A3 consists of 12 outcomes.

EXAMPLE 4.9   Table 4.1 gives information concerning 50 organ transplants in the state of Nebraska during a recent year. Each patient represented in Table 4.1 had only one transplant. If one of the 50 patient records is randomly selected, the probability that the patient had a heart transplant is = .30, since 15 of the patients had heart transplants. The probability that a randomly selected patient had to wait one year or more for the transplant is = .40, since 20 of the patients had to wait one year or more. The display in Table 4.1 is called a two-way table. It displays two different variables concerning the patients.

EXAMPLE 4.10   To find the probability of the event A that the sum of the numbers on the faces of a pair of dice equals seven when a pair of dice is rolled, consider the sample space shown in Fig. 4-4. The event A is the six points represented by large black squares connected by the grey line in Fig. 4-4. The sample space consists of the 36 points shown. The probability of the event A is 6/36 or 1/6.

The classical definition of probability is not always appropriate in computing probabilities of events. If a coin is bent, heads and tails are not equally likely outcomes. If a die has been loaded, each of the six faces do not have probability of occurrence equal to . For experiments not having equally likely outcomes, the relative frequency definition of probability is appropriate. The relative frequency definition of probability states that if an experiment is performed n times, and if event E occurs f times, then the probability of event E is given by formula (4.5).

P(E) =                                     (4.5)

EXAMPLE 4.11   A bent coin is tossed 50 times and a head appears on 35 of the tosses. The relative frequency definition of probability assigns the probability = .70 to the event that a head occurs when this coin is tossed. A loaded die is tossed 75 times and the face "6" appears 15 times in the 75 tosses. The relative frequency definition of probability assigns the probability = .20 to the event that the face "6" will appear when this die is tossed.

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