146 CHAPTER 4 Probability Three Common Approaches to Finding the Probability of an Event We first list some basic notation, then we present three common approaches to finding the probability of an event. Notation for Probabilities P denotes a probability. A, B, and C denote specific events. P1A2 denotes the “probability of event A occurring.” The following three approaches for finding probabilities result in values between 0 and 1: 0 … P1A2 … 1. Figure 4-2 shows the possible values of probabilities and the more familiar and common expressions of likelihood. 1. Relative Frequency Approximation of Probability Conduct (or observe) a procedure and count the number of times that event A occurs. P1A2 is then approximated as follows: P1A2 = number of times Aoccurred number of times the procedure was repeated When referring to relative frequency approximations of probabilities, this text will not distinguish between results that are exact probabilities and In the following display, we use “b” to denote a baby boy and “g” to denote a baby girl. Procedure Example of Event Sample Space: Complete List of Simple Events Single birth 1 girl (simple event) {b, g} 3 births 2 boys and 1 girl (bbg, bgb, and gbb are all simple events resulting in 2 boys and 1 girl) {bbb, bbg, bgb, bgg, gbb, gbg, ggb, ggg} Simple Events: ■ With one birth, the result of 1 girl is a simple event and the result of 1 boy is another simple event. They are individual simple events because they cannot be broken down any further. ■ With three births, the result of 2 girls followed by a boy (ggb) is a simple event. ■ When rolling a single die, the outcome of 5 is a simple event, but the outcome of an even number is not a simple event. Not a Simple Event: With three births, the event of “2 girls and 1 boy” is not a simple event because it can occur with these different simple events: ggb, gbg, bgg. Sample Space: With three births, the sample space consists of the eight different simple events listed in the lower right cell of the above table. CP YOUR TURN. Do Exercise 31 “Four Children.” EXAMPLE 2 Simple Events and Sample Spaces Certain Likely 50–50 Chance Unlikely Impossible 0 0.5 1 FIGURE 4-2 Possible Values for Probabilities C Probabilities That Challenge Intuition In certain cases, our subjective estimates of probability values are dramatically different from the actual probabilities. Here is a classical example: If you take a deep breath, there is better than a 99% chance that you will inhale a molecule that was exhaled in dying Caesar’s last breath. In that same morbid and unintuitive spirit, if Socrates’ fatal cup of hemlock was mostly water, then the next glass of water you drink will likely contain one of those same molecules. Here’s another, less morbid example that can be verified: In classes of 25 students, there is better than a 50% chance that at least 2 students will share the same birthday (day and month). I c s e p v m
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