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University of New South Wales
School of Actuarial Studies
ACTL 1001 Actuarial Studies and Commerce
Quiz 1 Solutions, 2004
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[10 marks]
Abraham de Moivre:
1. following the work of Huygens and Bernoulli, wrote on probability (
The
Doctrine of Chances
, 1718)—the Frst treatment of probability in En
glish.
2. applied the theory of probability to the problem of calculating the
value of an annuity (in his 1725 paper
Annuities upon Lives
). There
he assumed the number alive in the life table decreased in an arithmetic
progression.
3. probably inspired the work of Halley dealing with the problem of an
nuities which depend on the survival of two lives (joint life annuities).
4. Frst edition of
Doctrine of Chances
was dedicated to the great Isaac
Newton
5. de Moivre was buddies with Newton and Halley (the famous Astronomer
Royal)
Question 2
[30 marks]
We can represent our circumstances by Fgure 1.
(
a
) The chance that no die shows the number backed =
±
5
6
²
3
.
(
b
)”
o
n
e
d
i
e”
=
3
·
±
5
6
²
2
·
1
6
.
(
c
t
w
o
d
i
c
e
”
=
3
·
5
6
±
1
6
²
2
.
(
d
)
”
three dice
”
=
±
1
6
²
3
.
The net expectation of the player is
=(
b
)+2(
c
)+3(
d
)
−
(
a
)=
−
17
/
216
,
and that of the banker is
=
−
(
b
)
−
2(
c
)
−
3(
d
)+(
a
)=17
/
216
,
1
Gamble at t=0
1
1
2
3
Gains/Losses
Figure 1: Net payments from the game.
so the advantage lies with the banker in the long run.
For a stake of a dollar it is easily seen that this advantage is just under
eight cents.
Here it might be contended that as the player is to receive back his own
coin in addition to the prize, his expectation should be based on respective
receipts of two, three or four units instead of one, two or three units, as
appears in the above solution. His expectation on this basis might be argued
is being 2(
b
)+3(
c
)+4(
d
)
−
(
a
) or 74/216, which would show a substantial
advantage to the player. Further consideration would show, however, that
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This note was uploaded on 08/23/2010 for the course ACTL 1001 taught by Professor Bernartwong during the Three '10 term at University of New South Wales.
 Three '10
 BernartWong

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