CPS 102: Discrete Mathematics
Instructor:
Bruce Maggs
Assignment
3
Due:
Monday, October 24
th
, 2011
Note: You cannot discuss/collaborate with any other person other than the instructor or TA.
And, when solving for any question below please show your work!
1
Counting (5 points)
A college has 10 basketball players. A team of 5 players, of which 1 is a captain, will be selected
out of these 10 players. How many ways are there to form the team?
2
Probability (5 points)
What is the probability that a fivecard poker hand contains the ace of hearts?
3
Counting (5 points)
All PhD students in the CS department at Duke have to receive a
Quals Pass
in four of the seven des
ignated courses spread across three streams –
AI / Num. Analysis
:
{
CPS 250
,
CPS 270
,
CPS 271
}
,
Theory
:
{
CPS 230
,
CPS 240
}
and
Systems
:
{
CPS 210
,
CPS 220
}
. However, they cannot skip a
stream completely and they can only take one of
{
CPS 270
,
CPS 271
}
. In how many ways can the
PhD students choose four courses of the designated seven?
4
Probability (5 points)
What is the conditional probability that exactly four heads appear when a fair coin is flipped five
times, given that the first flip came up heads?
5
Counting (5 points)
How many bit strings of length 8 contain either four consecutive 0s or four consecutive 1s?
6
Probability (5 points)
Provide an example that shows that the variance of the sum of two random variables is not neces
sarily equal to the sum of their variances when the random variables are not independent.
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 Fall '09
 Variance, Probability theory, Bruce Maggs

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