Notes-08-23

Notes-08-23 - Introduction to Statistical Inference I Knowledge and Uncertainty Imagine the following game Unseen by you the Dealer places 19 beads

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Introduction to Statistical Inference I. Knowledge and Uncertainty Imagine the following game. Unseen by you, the Dealer places 19 beads of one color in 19 boxes and one bead of a diFerent color in a 20th box. The boxes, which all look identical from the outside, are closed and placed in a random arrangement on the table. You get to open one box and observe the color of the bead it contains. At this point, if you wish, you can make a wager about the dominant color. To do so, you put $20 on the table and announce your guess. The Dealer opens all the remaining boxes to reveal their contents. If you have guessed correctly, you take back your $20 and the Dealer pays you a dollar as a prize. If you are wrong, the Dealer takes your $20. Problems 1. Will you play this game? 2. Would you play if instead of risking $20, you had to put a diFerent amount down? ±or what amounts would you play? 3. This is a philosophical question. What do you know about the contents of the boxes after observing the contents of one box? Clearly, you know the rules of the game and you know the color of the bead in the box you opened. Do you know more than this? Solutions 1. Will you play this game? You should always bet that the dominant color is the color you’ve seen, since any other way of guessing will have a smaller chance of success. In the long run, in 19 out of 20 plays you will be right, resulting in a gain of $19. But in one out of 20 plays, you’ll lose $20. So, on average in each 20 plays, you’ll lose a dollar. If you are smart, you will not play. 2. Would you play if instead of risking $20, you had to put a diFerent amount down? ±or what amounts would you play? By the same reasoning as in the previous answer, if you put down $19 to play, then you will break even in the long run. If the amount you need to put down is any less than $19, the game is in you favor. 3. What do you know about the contents of the boxes after observing the contents of one box? What you know is very similar to what you know about random experiment (with a chance of success of 0.95) before the experiment in performed. You know enough to be able to make bets on the outcome in such a manner that the advantage will be yours in the long run. This simple example illustrates, in the simplest form I can think of, the pattern of rea-
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This note was uploaded on 11/29/2011 for the course MATH 4056 taught by Professor Staff during the Fall '08 term at LSU.

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Notes-08-23 - Introduction to Statistical Inference I Knowledge and Uncertainty Imagine the following game Unseen by you the Dealer places 19 beads

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