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MA 2631 Conference 1

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MA 2631 Conference 1
1. In how many ways can 3 science fiction books, 4 math books and 1 cooking book arranged on a
bookshelf, if
(a) there are no restrictions on the arrangement?
(b) all the science fiction books have to be stored together and also all the math books have to be
stored together?
(c) (only) the math books have to be stored together, the rest can be arranged without restriction?
2. Mike has nine friends and wants to throw a party. As his appartment is not big enough to invite all of
them, he decides to invite only six.
(a) How many choices of invitations has he?
(b) Two of his friends of his are feuding and will not attend the party together. Accounting for this
fact, how many possibilities has he?
(c) Two of his friends are very close and will attend the party only if invited together. Accounting
for this fact, how many possibilities has he?
3. A coin is tossed repeatedly until the first time ”heads” appears.
(a) Describe mathematically the sample space of this experiment.
(b) Describe mathematically the events
E = ”there are no more than four tails”
F = ”there are at least two tails”
(c) Describe mathematically the events E ∩ F and E ∪ F
c
4. Given a family of events E1, E2, . . . , En, . . . on some sample space Ω, construct a new family F1, F2, . . . , Fn, . . .
on the same sample space Ω such that the events Fi are monotone, (Fm ⊆ Fn for m ≤ n)and
[n
k=1
Fk =
[n
k=1
for any positive integer n.
Prove that the constructed family has the desired properties.
1

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