1.pdf - STATISTICS 244 Problem set 1 9:00-10:20 TTh Due...

Info icon This preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
STATISTICS 244 Problem set 1 9:00-10:20 TTh Due Thursday January 12, 2017 1. The French Lottery. Between April 1758 and May 1836 the French conducted a lottery very much like modern Lotto. Every two weeks there would be a Tirage (a draw) that consisted of a sequence of 5 integers drawn without replacement from 1, 2, 3, ..., 90. The result of a “typical” Tirage might look like this: 61, 4, 25, 1, 71. The rules for the French lottery differed slightly from modern Lotto: before the Tirage a player would select one, two, three, four, or five numbers between 1 and 90, and had a choice of bets. If the player selected a single number (called an “Extrait”), the player is betting that number will be among the five in the Tirage. If two numbers are specified (an “Ambe”) the bet is that the pair will be included in the Tirage. Similarly if three are specified (a “Terne”), four (a “Quaterne”), or five (a “Quine”). There was one further wrinkle in the betting: if the player bets an Extrait, there was a choice of two bets: a “Extrait simple” (the player would win if the number occurred in any of the five positions), or an “Extrait déterminé”, where the player would also specify in which of the five positions the number would occur. Similarly, an “Ambe déterminé” would have the player specify for each member of the pair exactly where in the five draws of the Tirage it would occur. For example, the following bets would have won with the above “typical” Tirage: (i) if the number 71 were bet as an Extrait simple, (ii) if the numbers 1, 25, 71 were bet as a Terne, (iii) if the numbers 1, 4, 25, 61, 71 were bet as a Quine, (iv) if the player bet the Ambe déterminé that the second number selected would be 4 and the third would be 25. Clearly the order in which the numbers are drawn only matters with “déterminé” bets. The payoffs would be a multiple of the bet placed, according to the following schedule. For example, if you bet $5 on the Extrait simple “54”, you would pay the lottery $5 at the time of the bet. You would lose the $5 if none of the five numbers in the Tirage were “54”, but if a “54” came up in any of the five positions you would receive $5x15=$75 (but they would keep the $5 you paid, so your net gain is $70). Type of Bet Payoff: Multiple of Amount Bet Extrait simple 15 Extrait déterminé 70 Ambe simple 270 Ambe déterminé 5,100 Terne 5,500 Quaterne 75,000 Quine 1,000,000 Assuming the Tirage is fairly conducted, (a). How many different ways can a Tirage be selected (without regard to order)? Compute the answer (i) exactly, with a hand calculator, (ii) approximately, using Stirling’s Formula (see Chapter 1 of Notes). (b) What is the expected value (following Huygens) of a single bet of $5 on a Terne? A Quaterne? (c) The above Payoff values are not favorable to the player. Calculate what the payoff multiples should be for the Extrait Simple, Terne, and Quaterne if Huygens were to consider the bets fair. [Hint: The probability of winning a bet on an Extrait déterminé is clearly 1/90, since there are 90 possible outcomes, only one of which is favorable. The payoff should
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern