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1 John is arranging his five math books and four statistics books on a shelf in a random order. How many different ordered arrangements of all nine books are possible? In how many of the arrangements will all statistics books end up clustered together? What is the probability that in John's random arrangement all statistics books will be clustered together?

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Introductory Statistics
OER Edition, 2013
Openstax
ISBN: 9781938168208

Related questions

Five different gifts will be distributed among four children. All gifts must be given, but it is not necessary for all children to receive a gift.

a. How many possible ways to distribute gifts are there if the above conditions are taken into account?
b. What is the probability that three children receive a gift?

2 (a) Suppose you start with one Red and one Black ball in a bag. You draw a ball at random from the bag repeatedly and look at its colour each time. After each successive draw, you double the number of Red balls and triple the number of Black balls in the bag before the next draw (eg. on the third draw there will be 4 Red and 9 Black balls in the bag). Explain why you will draw only finitely many Red balls in total with probability 1. [3 marks]

(b) In a modified game, suppose instead that for before the nth draw, you first make an n by n square arrangement of balls where the perimeter balls are all Red and any interior balls are Black ball. Thus, on the second draw there are 4 Red and 0 Black in the bag (22 in total), on the third draw there are 8 Red and 1 Black in the bag (32 in total), on the fourth there are 12 Red and 4 Black in the bag (42 in total), and so on. How many Red and Black balls will there be on the nth draw? Will you still draw only finitely many red balls? Explain your answer. [4 marks]

Since Thorndike “Graph-Pro” racquetball racquets were introduced several years ago, no attempt has ever been made to offer a lightweight version. Ted has done some research and found that some of the other manufacturers offer a lighter version of their standard racquet. In addition, some of the racquetball players he has talked to have said a lighter racquet would probably improve their game.

Talking with production engineers, Ted learns that the racquets coming off the line don’t all weigh the same anyway. According to the chief engineer, the racquets weigh an av- erage of 240 grams, but the weights “vary all over the lot,” and he takes advantage of the situation to press for a high-tech $700,000 machine that would produce racquets of equal quality while holding this variation within narrower limits.

Old Luke isn’t too keen on buying a $700,000 machine, especially since Ted might have a better solution. Ted has dusted off the literature that accompanied the racquet-production machine and found that the machine generates output that is approximately normally distributed, with a standard deviation of 10 grams. This is a little sloppy these days, but maybe Thorndike Sports Equipment can turn this weakness into a strength.

Conversing further with Luke, talking to three sporting goods retailers, and interviewing players at five different racquetball facilities, Ted comes back to the office with a terrific idea. Asking Luke to sit down, and bringing him some coffee, Ted begins his pitch: “Say, grandfather, why don’t we take the racquets at the lighter end of the range and label them ‘Graph-Pro Light,’ with the ones more in the middle being ‘Graph-Pro Regular,’ and the ones at the heavier end can be ‘Graph-Pro Stout’?”

Luke responds that just because this works for the beer companies, this doesn’t mean it will work for Thorndike Sports Equipment. However, he’s willing to give it a try. Af- ter accompanying Ted on another round of conversations with retailers and players, the elder Thorndike tells Ted that for now, he’d like to see 15% of the production consist of “Graph-Pro Light,” with 80% “Graph-Pro Regular,” and 5% “Graph-Pro Stout.”

Ted’s next task is to tell the production people how to go about selecting which racquets are to be put into these three different categories.

(a) What is the weight below which racquets classify as Graph-Pro Light? (b) What is the weight above which racquets classify as Graph-Pro Stout?

 

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