1. Knowing When to Stop: The Mathematics of Optimal Stopping 1 Minimize Cost and Maximize Reward

2. The Marriage Problem Suppose you decide to marry and interview at most 100 candidate spouses. The interviews are arranged in random order and you have no information about candidates you haven’t yet spoken to. After each interview you must either marry that person or forever lose the chance to do so. If you have not married after interviewing candidate 99, you must marry candidate 100. Your objective is to marry the absolute best candidate of the lot. How can you choose the best candidate? 2

3. You can select the very best spouse with probability at least 1/100 simply by marrying the first candidate, though you can do better Suppose you use 100 numbers, positive or negative and as large or small as possible to simulate the problem. Strategy #1 (i)Call an observed number a “record” if it is the highest number seen so far. (ii) Look at the first 50 numbers (or marriage candidates) and note any records. (iii) After that stop with the first record you see in the next 50 numbers. (iv)If the second highest number in 100 cards is in the first 50 you look at, and the highest is in the second half; you will select the highest number 1 in 4 times. 3

4. 4 First lot of 50 numbersSecond lot of 50 numbersWin or Lose Case 12 nd highest numberhighest numberwin Case 22 nd highest number and highest number lose Case 32 nd highest number and highest number lose Case 4Highest number2 nd highest numberlose Strategy #2 Observe 37 cards (or potential partners) without stopping, then stop with the next record. (This strategy guarantees stopping with the best number about 37% of the time. Probability of winning = N / e, where N = number of candidates and e = 2.718...

5. 5 John Elton (US Air Force Vietnam era) Asked friends to write down 100 numbers and bet them each $10 he could stop at the largest number. Convinced friends his chance of winning was very small. He would pay $1 to each person if he lost. Won one third of the time. (receiving $10 from each person)

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9. 9 Toss a coin repeatedly and stop whenever you want, receiving as a reward the average number of heads accrued at the time you stop. An Unsolved Problem

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11. 11 When Will My MP3 Player Repeat A Song? You have 1,000 songs on your MP3 player; how many songs must be played at random before the chances of a song being repeated is at least 50%? Birthday Paradox In a group of 23 people there is a 50% chance that two people will have the same birthday. (day and month)

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