obcourses - Obstacle Courses 1. Introduction. Here are two...

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Page 1 Obstacle Courses 1. Introduction. Here are two situations involving chance: (i) Three tickets will be taken out, one at a time, at random, from the box shown below, leaving three tickets behind in the box. (ii) A die will be rolled three times. In the F rst situation, it could happen the tickets come out: And in the second, it could happen the die comes up: Neither possibility is very likely. But do they have the same chance? Before reading on, you might look up from the page and try to answer the question. No experience in calculating chances is required, just reasoning. To get 1, 2, 3 from the box (in that order), the following must happen: The chances are not the same. The box is more likely to produce the 1, 2, 3 because on the second and third draws the chance of getting the right number is larger than the 1 in 6 chance for the die. 1 2 3 4 5 6 then F rst, then , 1 2 3 then F rst, then , the must be drawn from and then the must be drawn from and then the must be drawn from 1 2 3 4 5 6 2 3 4 5 6 3 4 5 6 1 3
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Page 2 The reasoning involved three small mental steps, each consisting of pinning down exactly what is in the box before another ticket is drawn. This kind of reasoning comes up often in chance calculations, and it helps to have a diagram to guide you through the steps. The idea behind the diagram is as follows. Think of the box and tickets as hoping they are going to produce the 1, 2, 3: You know three things stand in their way. On the F rst draw, the 1 must come out of the box. If another number shows up, the box and tickets will have their hopes dashed. So that makes one obstacle: After that, the box and tickets face another obstacle. The next ticket must be a 2: And there is one obstacle left: ±inally, the diagram shows the chances: ±or example, the number under the second obstacle is 1/5. ±or the box and tickets to get over that obstacle, the 2 has to be drawn from 2, 3, 4, 5, 6 and the chance of that is one in F ve or 1/5. The diagram is an obstacle course. Obstacle courses are useful when a 1 F rst 2 next 1 F rst 1 F rst 1/6 2 next 1/5 3 last 1/4 1 F rst 2 next 3 last 1 2 3 4 5 6 We all want then then ! 3
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Page 3 chance situation proceeds in stages, and you want to know the chance of something happening at the F rst stage and something else happen- ing at the second stage and something else again at the third stage, and so on. In the example, there were 3 stages—the three draws from the box—and the chance being considered is that of getting 1 at the F rst stage and 2 at the second and 3 at the third. With the exception of the F rst example, the examples below show you how to set up an obstacle course. The F rst example just reviews some basic chances for a deck of cards. If you haven’t played any card games for a while, you might want to read the following before going on. A deck of playing cards contains 52 cards:
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obcourses - Obstacle Courses 1. Introduction. Here are two...

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