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) Someone rolls a die three times. (People usually roll dice in pairs, so “dice” is more common than “die”, the singular form.) (ii) A box contains six tickets marked as shown: 1 2 3 4 5 6 Someone takes out three tickets, one at a time, at random, from the box, leaving three tickets behind in the box. In (i), it could happen the die comes up: then Frst, then , And in (ii), it could happen the tickets come out: then Frst, then , 1 2 3 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, only a little reasoning.
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Page 2 To get 1, 2, 3 from the box (in that order), the following must happen: the must be drawn from and then the must be drawn from and then the must be drawn from 2 3 4 5 6 1 2 3 4 5 6 3 4 5 6 1 2 3 The chances are not the same. The box is more likely than the die to lead to the 1, 2, 3 because on the second and third draws, the chance of getting the right number from the box is larger than the one in six chance of rolling the right number with the die. Part of the reasoning involved three small mental steps, each consist- ing 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: We all want then 1 2 3 then ! 1 2 3 4 5 6 You know three things stand in their way. On the Frst 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: Frst 1 After that, the box and tickets face another obstacle. The next ticket must be a 2: Frst 1 next 2
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And there is one obstacle left: Frst 1 next 2 last 3 That does part of the diagram. To Fnish it, you have to write a chance under each obstacle. Take the obstacles one at a time. The Frst one is straightforward. You want to Fnd the chance the box and tickets make it over the Frst obstacle. ±or that to happen, the 1 must be drawn Frst, and the chance of that is 1/6. Write that under the Frst obstacle: 1/6 Frst 1 next 2 last 3 The second obstacle will take more effort. ±irst—and this is the key step in arriving at the correct chance—imagine that the box and tickets made it over the Frst obstacle. In this example, if the box and tickets got over the Frst obstacle, then the 1 is no longer in the box, leaving the other Fve tickets for the seond draw: 2 3 4 5 6 Now you are ready to Fnd the chance for the second obstacle. The box and tickets made it over the Frst obstacle, and now they want to get over the second one. And for that to happen, the 2 must come up: We hope we 2 next. get 2 3 4 5 6 The chance is 1/5. Put that fraction under the second obstacle:
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obcourses - Obstacle Courses 1. Introduction. Here are two...

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