Stat 24400 Homework 1 Solution
Jan 12, 2017
Total points: 100
1.
[20pts] The French Lottery
(a) A Tirage is selected by choosing 5 numbers from 90. Therefore, the number of
Tirages that can be selected is
i.
✓
90
5
◆
= 43949268
.
ii. Use Stirling’s formula,
✓
90
5
◆
⇡
1
p
2
⇡
·
90
✓
1

5
90
◆

(90

5+1
/
2)
✓
5
90
◆

(5+1
/
2)
⇡
44689346
.
(b) In order to win your Terne bet, the lottery must choose your three numbers and
two numbers from the remaining 87; for the Quaterne, your four numbers and one
from the remaining 86. So if you pick
k
numbers, your probability of winning is
p
k
=
P
(win with
k
#
0
s
) =
(
k
k
)(
90

k
5

k
)
(
90
5
)
.
You win $5 times the multiple with probability
p
k
and $0 otherwise. Therefore
the expected payo
↵
of the bets are
Terne:
p
3
·
5500
·
$5 = $2
.
34
.
Quaterne:
p
4
·
75000
·
$5 = $0
.
73
.
Alternatively
, the expected net gain are
Terne:
p
3
·
(5500)
·
$5

$5 =

$2
.
66
.
Quaterne:
p
4
·
(75000)
·
$5

$5 =

$4
.
27
.
(c) From the hint, in order for the bet to be fair, the payo
↵
must equal 1
/P
(win). We
already calculated the probability of winning the Simple bets in part (b). Now
we calculate the probabilities for the Determine bets. The Extrait is given in the
hint. Without loss of generality, suppose that in the Ambe bet, you predicted your
numbers would come in positions one and two in the Tirage. In order to win, the
lottery must select your number in the first position, which has probability 1
/
90.
Given that the first number matches, the probability that the second number
matches is 1
/
89, so the probability of winning is
1
90
1
89
. The following table gives
the resulting fair payo
↵
s.
1
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Type of Bet
Fair Payo
↵
/Fair Net Gain
Actual Payo
↵
Extrait Simple
18/17
15
Extrait Determine
90/89
70
Ambe Simple
400.5/399.5
270
Ambe Determine
8,010/8,009
5,100
Terne
11,748/11,747
5,500
Quaterne
511,038/511,037
75,000
Quine
43,949,268/43,949,267
1,000,000
(d)
i. The event that you win at least one of the bets is the complement of the event
that you win zero of the bets, meaning that you lose all of the bets. Therefore,
the probability of winning at least one of the bets is 1

P
(lose all bets). In
order to lose all of the bets, none of your numbers may appear in the Tirage.
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