Math 361, Problem set 4
Due 9/27/10
1. (1.4.26) Person
A
tosses a coin and then person
B
rolls a die. This is re
peated independently until a head or one of the numbers 1
,
2
,
3
,
4 appears,
at which time the game is stopped. Person
A
wins with the head, and
B
wins with one of the numbers 1
,
2
,
3
,
4. Compute the probability
A
wins
the game.
Answer:
The probability
A
wins on his
i
th coin flip is
1
2
(
1
6
)
i

1
; as he must
flip a head on the
i
th flip, and all other turns he must flip a tail while
player
B
rolls a 4 or 5. If
A
is the event
A
wins then
P
(
A
) =
∞
i
=1
1
2
1
6
i

1
=
1
/
2
5
/
6
=
3
5
.
A clever alternate method is the following. Suppose we consider the turn
when either
A
or
B
first wins.
Then
A
wins if and only if he flipped a
head on that turn. Let
A
be the event player
A
rolls a head that turn,
and
B
be the event player
B
rolls a 4 or 5. We want
P
(
A

A
∪
B
) =
P
(
A
)
P
(
A
∪
B
)
=
1
/
2
1

1
/
6
=
3
5
.
Note that this trick doesn’t work so well to calculate the probability that
B
wins. The reason is that
B
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 Fall '10
 Dr.PaulHorn
 Px, ith coin ﬂip

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