University
Solutions to Second Hour Exam
ECE 313
of Illinois
Fall 1997
1.
An experiment consists of tossing a fair coin repeatedly until a Head occurs for the tenth
time.
(a)
Suppose that the tenth Head occurred on the twentyfifth toss.
What is the conditional
probability that the first toss resulted in a Head?
(b)
Let the random variable
X
denote the number of Tails that occur before the tenth Head
occurs.
For example, if the tenth Head occurred on the twentyfifth toss as was assumed
in part (a), then
X
= 15 for this particular outcome of the experiment.
More generally,
X
can take on any nonnegative integer value 0, 1, 2, 3, …
.
What is P{
X
= k}?
1.(a)
P{first toss is a Head

tenth Head on twentyfifth toss}
=
P{first toss is a Head
∩
tenth Head on twentyfifth toss}
P{tenth Head on twentyfifth toss}
=
P{H; 8 Heads in 23 tosses; H}
P{tenth Head on twentyfifth toss}
=
()
1
2
23
8
1
2
8
1
2
23–8
1
2
24
9
1
2
25
=
23
8
24
9
=
23!9!15!
24!8!15!
=
9
24
=
3
8
.
Alternatively, given that the tenth Head was on the 25th toss, we know
for sure that nine Heads occurred
on 24 tosses.
Thus, the probability that the first toss resulted in a Head is 9/24 according to a formula
derived in class.
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 Spring '08
 Sarwate
 Probability theory, Discrete probability distribution, odd number, twentyfifth toss, Head tenth Head

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