chance_variables

chance_variables - Chance Variables 1. Chance variables....

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Page 1 Chance Variables 1. Chance variables. When a pair of dice are rolled, the total number of points that show could be either 2, 3, … 11, or 12. When a coin is tossed 10 times, the number of times a head comes up could be 0, 1, … 9, or 10. The total number of points, and the number of times a head comes up, are examples of chance variables . In the Frst example, the numbers 2 through 12 are said to be the possible values of the total number of points. In the second example, the numbers 0 through 10 are the pos- sible values of the number of times heads comes up. People often drop the “possible”, and just use “the values”. In mathematics, entities that can have different values within the same context are often referred to as variables. The total number of points (when a pair of dice are rolled) is a variable because it is possible for it to be any number between 2 and 12 inclusive. It is a “chance” variable because the possible values have chances: the value 2 has chance 1/36, the value 3 has chance 2/36, and so on. Example 1. If a roulette player stakes $1 on red at roulette, two things can hap- pen. Either he wins, in which case the casino returns his stake along with another dollar as prize; or he loses, in which case the casino keeps the stake. A roulette player bets $1 on red six times in a row. His net gain from the six bets is deFned as: net gain = amount he wins — amount he loses Net gain is a chance variable. What are its possible values? Answer. What is the best that could happen to the gambler? He could win ev- ery time. In that case his net gain would 6 (dollars). What is the worst that could happen? He could lose every time, and the net gain would be –6 (dollars). So the possible values are whole numbers between –6 and 6 (inclusive). But not all whole numbers. Suppose the gambler wins Fve times and loses once. His net gain would be 4 (dollars). The possible values of net gain are: –6, –4, –2, 0, 2, 4, 6.
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Page 2 2. Distribution table of a chance variable. The distribution table of a chance variable is a table with two col- umns. The Frst column lists the possible values of the variable. The second column displays, beside each value in the Frst column, the chance the variable turns out to be that value. Example 2. A die is rolled twice. The number of times the single dot ( )shows up in the two rolls is a chance variable. ±ind its distribution table. Answer. The Frst step is to get the values of the chance variable. The single dot could show up on neither roll, one of the rolls, or both. So the pos- sible values are 0, 1, 2. That gives the Frst column of the distribution table: value 0 1 2 To get the second column, you have to work out the three chances below: value 0 chance the single dot shows up 0 times 1 chance the single dot shows up 1 time 2 chance the single dot shows up 2 times That leads to the following table: value chance 0 25/36 1 10/36 2 1/36 This is the distribution table for the number of times
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chance_variables - Chance Variables 1. Chance variables....

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