148 CHAPTER 4 Probability Simulations Sometimes none of the preceding three approaches can be used. A simulation of a procedure is a process that behaves in the same ways as the procedure itself so that similar results are produced. Probabilities can sometimes be found by using a simulation. See Section 4-5. ROUNDING PROBABILITIES It is difficult to provide a universal rule for rounding probability values, but this guide will apply to most problems in this text: When expressing the value of a probability, use this rounding rule: Either give the exact fraction or decimal, or round off final decimal results to three significant digits. (Suggestion: When a probability is not a simple fraction such as 2>3 or 5>9, express it as a decimal so that the number can be better understood.) All digits in a number are significant except for the zeros that are included for proper placement of the decimal point. See the following examples. ■ The probability of 0.1827259111333 (from Example 6) has thirteen significant digits (1827259111333), and it can be rounded to three significant digits as 0.183. ■ The probability of 1>3 can be left as a fraction or rounded to 0.333. (Do not round to 0.3, because 0.3 has only one significant digit instead of three.) ■ The probability of 2>8 can be expressed as 1>4 or 0.25. (Because 0.25 is exact, there’s no need to express it with three significant digits as 0.250.) Probabilities Expressed as Percentages? Mathematically, a probability of 0.25 is equivalent to 25%, but there are good reasons for working with fractions and decimals and not using percentages. Professional journals almost universally express probabilities as decimals, not as percentages. Later in this book, we will use probability values generated from statistical software, and they will always be in the form of decimals. Also, calculations using probability values, such as 0.25 * 0.25, are easy when working with decimals, but 25,* 25, can cause major problems. When finding probabilities with the relative frequency approach, we obtain an approximation instead of an exact value. As the total number of observations increases, the corresponding approximations tend to get closer to the actual probability. This property is commonly referred to as the law of large numbers. LAW OF LARGE NUMBERS As a procedure is repeated again and again, the relative frequency probability of an event tends to approach the actual probability. The law of large numbers tells us that relative frequency approximations tend to get better with more observations. This law reflects a simple notion supported by common sense: A probability estimate based on only a few trials can be off by a substantial amount, but with a very large number of trials, the estimate tends to be much more accurate. S la so si Understanding Chances of Winning the Lottery In the New York State Lottery Mega Millions game, you must choose five different numbers from 1 to 75, and you must also select another “Mega Ball” number from 1 to 15. To win the jackpot, you must get the correct five numbers and the correct Mega Ball number. The chance of winning the jackpot with one ticket is 1>258,890,850. Commercials for this lottery state that “all you need is a little bit of luck,” but in reality you need a ginormous amount of luck. The probability of 1>258,890,850 is not so easy to understand, so let’s consider a helpful analogy suggested by Brother Donald Kelly of Marist College. A stack of 258,890,850 quarters is about 282 miles high. Commercial jets typically fly about 7 miles high, so this stack of quarters is about 40 times taller than the height of a commercial jet when it is at cruising altitude. The chance of winning the Mega Millions lottery game is equivalent to the chance of randomly selecting one specific quarter from that pile of quarters that is 282 miles high. Any of us who spend money on this lottery should understand that the chance of winning the jackpot is very, very, very close to zero. In S M g c fe f CAUTIONS 1. The law of large numbers applies to behavior over a large number of trials, and it does not apply to any one individual outcome. Gamblers sometimes foolishly lose large sums of money by incorrectly thinking that a string of losses increases the chances of a win on the next bet, or that a string of wins is likely to continue. 2. If we know nothing about the likelihood of different possible outcomes, we should not assume that they are equally likely. For example, we should not think that the probability of passing the next statistics test is 1>2, or 0.5 (because we either pass the test or do not). The actual probability depends on factors such as the amount of preparation and the difficulty of the test.
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