STAT 401 Lab 2
5/30/2013
Just as a reminder, the web page is http://gzt.public.iastate.edu/
stat401/ and my e-mail address is gzt@iastate.edu. I don’t have scheduled
office hours, but if you want to discuss questions feel free to e-mail me to set up
a time.
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HW Questions?
Note to ask the instructor questions about HW problems, lecture examples,
book examples, or other material you don’t understand.
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Example Problems
2.1
Efron Dice
I have four dice, A, B, C, D. They are fair dice (each face of each die will come
up with probability 1/6. The faces are labeled as follows:
A: 4 4 4 4 0 0
B: 3 3 3 3 3 3
C: 6 6 2 2 2 2
D: 5 5 5 1 1 1
So four faces of die A are labeled 4. Therefore, e.g.:
P (A = 4) = 4(1/6) = 2/3
Find the following:
• P (A > B)
• P (B > C)
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• P (C > D)
Note: this one is tricky.
Idea: if I roll die C and get a 2, what is the probability that I beat D?
Continue for all possibilities.
• P (D > A)
Note: same method as the previous.
• OPTIONAL CHALLENGE: you get to choose one die and somebody else
will randomly select one of the three remaining dice. Whoever then rolls
the highest number wins a giant sack of cash. Which die should you
choose?
These dice behave in a counterintuitive manner!
2.2
Normal Dice
We are now using normal, 6-sided dice, X.
That is, there are 6 sides, each side is labeled with a different integer between
1 and 6, and each side has probability 1/6 of being rolled.
• Sketch the pmf and cdf of X.
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• If you roll one die, X, what is the mean of X?
• What is the variance of X?
• What is the mean of 2X?
• What is the variance of 2X?
• I’m playing a board game. I have a choice between either rolling two dice,
X1 and X2 , and adding them up or rolling one die, X, and using 2X for
my turn. It’s the last turn of the game. If I get a 10 or higher, I win.
Otherwise, I lose. Should I roll 2 dice or roll one die and double it?
• What about if I only need a 7 or higher to win?
• Okay, this one isn’t a normal die anymore: I have an 8-sided die labeled
from 0 to 7. What is the mean and what is the variance of its rolls?
2.3
Coloring a Complete Graph Randomly
A complete graph G on n vertices is a graph with an edge connecting each pair
of vertices.
Insert crude doodle:
There are n2 = n(n−1)
edges in G.
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Suppose each edge is colored red with probability p and blue with probability
q = 1 − p. Every edge is either colored red or blue.
• What is the expected number of red edges in G?
• What is the standard deviation of the number of red edges in G?
• Suppose n > 3, 0 < p < 1. If I choose 3 vertices at random from G, what
is the probability that the edges connecting them to each other are all
red? What is the probability that they are all blue?
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• TRICKY AND OPTIONAL: If I choose k vertices, 1 < k < n, what is the
probability that all of the edges are one color? e.g. all red or all blue.
• Let n = 20 and k = 5. That is, we choose 5 vertices at random from the
graph with their edges and call this new graph K.
Suppose the number of red edges in G is 50.
What is the probability that there are exactly 2 red edges in K?
HINT: hypergeometric.
• If we use the binomial approximation, what is our estimate of the probability that there are exactly 2 edges in K?
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References
• Efron Dice
• Complete graph
• STAT 401 Page
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