4-1 Basic Concepts of Probability 149 Relative Frequency Probability: Airline Crashes EXAMPLE 3 Find the probability that a commercial airliner will crash on any given flight. YOUR TURN. Do Exercise 21 “YSORT Gender Selection.” SOLUTION In a recent year, there were about 39 million commercial airline flights, and 16 of them crashed. We use the relative frequency approach as follows: P1airline crash2 = number of airline crashes total number of airline flights = 16 39,000,000 = 0.000000410 Here, the classical approach cannot be used because the two outcomes (crashing, not crashing) are not equally likely. A subjective probability can be estimated in the absence of historical data. Classical Probability: Three Children of the Same Gender EXAMPLE 4 When three children are born, the sample space of genders is as shown in Example 2: {bbb, bbg, bgb, bgg, gbb, gbg, ggb, ggg}. If boys and girls are equally likely, then those eight simple events are equally likely. Assuming that boys and girls are equally likely, find the probability of getting three children all of the same gender when three children are born. (In reality, the probability of a boy is 0.512 instead of 0.5.) YOUR TURN. Do Exercise 29 “Three Children.” SOLUTION The sample space {bbb, bbg, bgb, bgg, gbb, gbg, ggb, ggg} includes eight equally likely outcomes, and there are exactly two outcomes in which the three children are of the same gender: bbb and ggg. We can use the classical approach to get P1three children of the same gender2 = 2 8 = 1 4 or 0.25 SOLUTION Earning a grade of A depends on many factors including these: How much you study; the difficulty of the tests; attendance in class; best in class textbook (✓); skill of your professor. There is no way to calculate an accurate probability, so we use subjective judgment. You might observe that you enjoy learning statistical concepts and are highly motivated to master this valuable subject which has countless applications in today’s data-driven world! This passion for statistics and your success in previous math courses might all suggest that there is a good chance of earning an A, so you might assign a probability of 0.8 to that event. Subjective Probability: Earning a grade of A in this Statistics Course EXAMPLE 5 What is the probability that you will earn a grade of A in the statistics course you are currently taking? YOUR TURN. Do Exercise 4 “Subjective Probability.” How Probable? How do we interpret such terms as probable, improbable, or extremely improbable? The Federal Aviation Administration (FAA) interprets these terms as follows. • Probable: A probability on the order of 0.00001 or greater for each hour of flight. Such events are expected to occur several times during the operational life of each airplane. • Improbable: A probability on the order of 0.00001 or less. Such events are not expected to occur during the total operational life of a single airplane of a particular type, but may occur during the total operational life of all airplanes of a particular type. • Extremely improbable: A probability on the order of 0.000000001 or less. Such events are so unlikely that they need not be considered to ever occur.
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