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Math 135, Fall 2011: HW 6
This homework is due at the beginning of class on
Wednesday October 19th, 2011
. You are
free to talk with each other and get help. However, you should write up your own solutions and
understand everything that you write.
Problem 1
(p.217 #1)
.
A coin which lands heads with probability
p
is tossed repeatedly. Assuming
independence of the tosses, ﬁnd formulae for
a) the probability that exactly 5 heads appear in the ﬁrst 9 tosses;
b) the probability that the ﬁrst head appears on the 7th toss;
c) the probability that the ﬁfth head appears on the 12th toss;
d) the probability that the same number of heads appear in the ﬁrst 8 tosses as in the next 5
tosses.
Part (a) is a binomial (n,p) probability, where
n
= 9. Let
q
= 1

p
. We get
P
(exactly 5 heads in ﬁrst 9 tosses) =
±
9
5
²
p
5
q
4
.
Part (b) is a geometric distribution; in order for the ﬁrst head to appear at the 7th toss, the previous
6 tosses must be tails and the 7th toss must be heads. By independence, this is
q
6
p
.
Part (c) is a negative binomial distribution. In order for the ﬁfth head to appear on the 12th toss,
the 12th toss must be a heads, and there must be exactly 4 heads in the previous 11 tosses. Thus
P
(5th head appears on 12th toss) =
±
11
4
²
p
4
q
7
p
=
±
11
4
²
p
5
q
7
.
For part (d), observe that by independence of trials, the random variables
X
1
= [number of heads in the ﬁrst 8 tosses]
X
2
= [number of heads in the next 5 tosses]
are independent. Also
X
1
∼
B
(8
,p
), and
X
2
∼
B
(5
,p
). Hence
P
(same # of H in ﬁrst 8 as in next 5 tosses) =
5
X
k
=1
P
(
X
1
=
k,X
2
=
k
)
=
5
X
k
=1
P
(
X
1
=
k
)
P
(
X
2
=
k
)
=
5
X
k
=1
±
8
k
²
p
k
q
8

k
±
5
k
²
p
k
q
5

k
Problem 2
(p.218 #4)
.
A game is played in which three people each toss a fair coin to see if one
of their coins shows a diﬀerent face from the other two. The game ends if one of the three people
has a coin which lands diﬀerently from the other two.
1
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View Full Documenta) What is the probability of the game ending after one round of play?
b) What is the probability that the game lasts
r
rounds?
c) What is the expected duration of play?
SOLUTION. The probability of one of the coins landing diﬀerently from the other two is the com
plement of the probability that all three land heads or all three land tails. So
P
(game ends after one round) = 1

±
1
8
+
1
8
²
=
3
4
.
The probability the game ends after one round is
3
4
.
If game ends after
r
rounds, the ﬁrst
r

1 rounds must have ended with no coin landing diﬀer
ently from the others, and one coin landing diﬀerently from the other two on the
rth
round. By
independence, this is
P
(game lasts r rounds) =
±
1
4
²
r

1
±
3
4
²
.
Since the number of rounds of play is a geometric random variable with success probability
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 Fall '08
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 Math

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