breakingdown

breakingdown - Breaking Things Down 1 Introduction By...

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Page 1 Breaking Things Down 1. Introduction. By itself, the method of obstacles only works in a narrow range of cases. For example, suppose two tickets will be drawn, one at a time, at random from the box shown below: A A Z Z If you want to ± nd the chance the same letter shows on both tickets, the method of obstacles can handle this easily. The ± rst ticket can be any ticket but the second must have the same letter as the ± rst one: any ticket 4/4 same letter as ± rst 1/3 The answer is 1/3. Now suppose you want to ± nd the same chance with a slightly different box: A A Z Z Z You can’t use an obstacle course, because the chance the second let- ter is the same as the ± rst could be 1/4 or 1/2, depending on whether
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Page 2 the letter on the f rst ticket is A or Z. Something else is required. But beFore reading on, you might try to guess which box gives a better chance oF getting the same letter twice: the Four ticket box or the f ve ticket box. Here is the method. You break down the possibility: same letter shows on both tickets into diFFerent ways it can happen: get A on both tickets or get Z on both tickets Next, you f nd the chances For these (using obstacles): get A on both tickets: 2/5 × 1/4 = 2/20 or get Z on both tickets: 3/5 × 2/4 = 6/20 And the last step is to add the chances oF the diFFerent ways: 2/20 + 6/20 = 8/20 = 2/5. That ends the method. It leads to a 2/5 chance the same letter will come up twice. This is bigger than the chance For the Four ticket box, which is 1/3. (You might have guessed the chance would go up by imagining an extreme case: that the box contained 2 As and 100 Zs, say. Then the chance would be almost certain both letters would be the same, because they would both be very likely to be Zs. So even adding a single Z to the box will put the chance up.) Example 1. Someone shuFfl es a deck oF cards and deals out, one at a time, f ve cards. ±ind the chance there is not more than one ace in the f ve cards. Do you think the chance will turn out to be closest to 50%, 75%, or 100%? Answer. ±irst, break down the possibility: not more than one ace as Follows. There won’t be more than one ace iF either: there are 0 aces in the f ve cards or there is 1 ace in the f ve cards There are two chances to f nd. The f rst one is straightForward, be-
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Page 3 cause 0 aces in f ve cards can be written as an obstacle course: 1st not ace 48/52 2nd not ace 47/51 3rd not ace 46/50 4th not ace 45/49 5th not ace 44/48 The chance there are 0 aces in the f ve cards: 48/52 × 47/51 × 46/50 × 45/49 × 44/48 ≈ 0.6588 That does the f rst line oF the breakdown: not more than one ace in the f ve cards iF either: there are 0 aces in the f ve cards: 0.6588 or there is 1 ace in the f ve cards The second line takes most of the work. The possibility oF 1 ace (in the f
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This note was uploaded on 02/05/2011 for the course STAT 134 taught by Professor Aldous during the Spring '03 term at Berkeley.

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breakingdown - Breaking Things Down 1 Introduction By...

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