probability

Risk-Free Betting on Different Beliefs

Alice believes that the 49ers will win the Super Bowl with probability 5/8. Bob believes that Ravens will win the Super Bowl with probability 3/4. Assuming that Alice and Bob are both willing to accept any bet that gives them a positive expected value of...

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Medical Tests and Bayes’ Theorem

Suppose that you are worried that you might have a rare disease. You decide to get tested, and suppose that the testing methods for this disease are correct 99 percent of the time (in other words, if you have the disease, it shows that you...

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St. Petersburg Paradox

A new casino offers the following game: you toss a coin until it comes up heads. If the first heads shows on the N-th toss, you win 2N dollars. (Thus the payoff doubles with each coin toss that isn’t heads.) How much should you be willing...

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Farmers and Pesky Birds

Alice and Bob are two farmers each wanting to plant a (countably infinite) row of seeds, side by side in a field. Both of them have pesky birds that hinder their efforts in funny ways. As Alice walks along the row, sequentially dropping her seeds,...

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Poker Probability Paradox

In poker, there is a hierarchy of desirable hands which relate to how unlikely the hand is. For instance, a royal flush is deemed most desirable because it is least likely to arise in play. But what if you allow wild cards? Clearly this creates...

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Monty Hall Problem

Here’s a problem that makes the round every few years, and each time, it is hotly debated. You are on a game show. You are presented with a choice of 3 doors: behind one is a luxury car, and behind the other two are nothing....

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Deal or No Deal

On the TV show “Deal or No Deal”, a contestant is faced with a number of briefcases (26 in the US version) in which various amounts of money have been placed. The contestant is asked to select one to keep, unopened. Then the player plays...

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Seven Shuffles

How many shuffles does it take to randomize a deck of cards? The answer, of course, depends on what kind of shuffle you consider. Two popular kinds of shuffles are the random riffle shuffle and the overhand shuffle. The random riffle shuffle is modeled by...

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Pass the Beer

Suppose you and a bunch of friends are sitting around a table. There are N of you. You have a jug of beer in front of you, which no one has yet tasted. So you take a swig of it, and then pass it to...

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Two Envelopes Paradox

I have two envelopes, and inside each I have put some money. In fact, one envelope contains twice as much money as the other. I’ll let you select one envelope, which you can have after the game is over. But as soon as you select...

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