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# Assignment1-Stat230-Fall12.pdf

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University of Waterloo

Statistics

STAT 230

Kamyar Ghavam

Fall

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1
Assignment 1, STAT230-Fall 2012
Due on Wed. Sept. 26, 2012
The ﬁrst three questions consider the process of arranging coloured marbles in a row from left to right.
Two marbles of the same colour are to be considered indistinguishable when counting arrangements.
1. (a) Suppose that there are 4 white and 2 black marbles. List all 15 ways to arrange these marbles.
(b) What combinatorial number describes the number of ways to arrange n black and 4 white marbles?
Check your answer for the case n = 2.
(c) In which of the arrangements in 1a (list them) is every white marble adjacent to at least one other
white marble?
(d) List the ways to arrange 2 black and 2 red marbles.
(e) Explain why the lists in 1c and 1d have the same size? (Hint: Consider replacing each red marble
with two consecutive white marbles.)
(f) How many ways are there to arrange n black and 4 white marbles so that every white marble is
adjacent to at least one other white marble? (Check that your answer agrees with part 1c.)
(g) If 22 black and 4 white marbles are arranged at random, what is the probability that every white
marble is adjacent to at least one other white marble?
2. (a) Suppose that there are 2 black and 5 white marbles. List all 21 ways to arrange these marbles.
(b) What combinatorial number describes the number of ways to arrange n black and 5 white marbles?
Check your answer for the case n = 2.
(c) In which of the arrangements in 2a (list them) is every white marble adjacent to at least one other
white marble? (Note: There are nine such arrangements.)
(d) i. List the ways to arrange 2 black, 1 red, and 1 blue marble.
ii. If each blue marble in the preceding list is replaced by 3 white marbles and then each red marble
is replaced by 2 white marbles, then every arrangements from 2c will be listed, but some will be listed
twice. Describe which arrangements are listed twice.
(e) How many ways are there to arrange n black and 5 white marbles so that every white marble is
adjacent to at least one other white marble? (Check that your answer agrees with part 2c.)
(f) If 21 black and 5 white marbles are arranged at random, what is the probability that every white 2
marble is adjacent to at least one other white marble?
3. (Difﬁcult) If 5 black, 7 red, 9 blue, and 6 white marbles are arranged at random, what is the probability
that every white marble is adjacent to at least one other white marble?
4. Suppose that A and B are two events. Draw a Venn diagram and use it to explain how P

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