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