April 21, 2005
Physics 681481; CS 483: Discussion of #6
1.
The probability that you get the marked item on the first attempt is
1
4
.
The
probability that you get the marked item on the second attempt is also
1
4
: the probability,
3
4
, that you fail on the first attempt times the probability,
1
3
that you succeed on the
second. The probability that you have to make a third attempt is
1
2
: the probability,
3
4
that you fail on the first attempt times the probability
2
3
that you fail on the second. You
never need to make a fourth attempt, since if the third attempt also does not produce
the marked item you know it is the fourth.
Therefore the mean number of attempts is
1
4
×
1 +
1
4
×
2 +
1
2
×
3 = 2
1
4
.
Alternatively, note that you can do no better than to try out the four items in a
completely random order, since you have no information at the start, and failure at any
stage teaches you nothing about which of the remaining items is likely to be the marked
one. So whatever order you pick the items in, it is equally likely that the marked item will
be the first, second, third, or fourth on the list. Since the probability is
1
4
that the marked
item is first on the list,
1
4
that it is second, and
1
4
+
1
4
=
1
2
that it is third or fourth, the
mean number of attempts is indeed
1
4
×
1 +
1
4
×
2 +
1
2
×
3 = 2
1
4
.
2.
The action of the various
U
f
gates on the computational basis is as follows: In
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 Spring '05
 ANON
 Physics, Rightwing politics, Political spectrum, The Gates, Christo and JeanneClaude, central park, Toffoli gate

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