Hints for Homework Assignment 3
General remarks
Most problems fall into the success/failure framework and should be done
in the same way as the class examples. Be sure to say what you mean by
"trial" and by "success", and to specify the parameters n and p, when using
a success/failure trial model.
The majority of these problems (but not
all!), are of one of the two special types discussed in class, i.e., they
can be phrased in terms of probabilities involving the number of successes
occurring in a given number of S/F trials, or the number of the trial
of the first sucess. As with all word
problems, it is essential that you read the problems carefully, in order
to correctly identify the event in question. If you are not certain,
start writing down explicitly the S/F sequences that make up this event,
and some sequences that are not in the event; this may give you an idea
of how to describe the event in terms of successes and failures.
One problem does not fall into one of the two special cases
and therefore must be solved by "assembly/machine level" computations.
This means that you have to identify
the event in question explicitly as a set of certain
S/F sequences, and compute the probability of that event by
adding up the probabilities for all sequences in that set.
Specific Comments
- Problem 1:
This is a review problem, similar to the lottery problem discussed
in class a few weeks ago (to illustrate "sampling with replacement" versus
"sampling without replacement").
The problem requires constructing, for each of the two parts (a) and (b),
an appropriate Omega, identifying the event in question with a
subset of Omega, and computing its probability via the counting formula
P(A)=#(A)/#(Omega).
- Problem 2:
As stated on the problem sheet, this problem is basically a set-theoretic
exercise, and you have define appropriate sets (events) and draw Venn
diagrams.
- Problem 3:
This requires the assembly level probability computations
alluded to above.
Write down all S/F sequences that make up the event in question,
compute the probability for each of these sequences, and
add up these probabilities to get the probability for the
entire event.
(Note that saying "getting two sixes in a row" does not
exclude the possibility of getting
a longer string of sixes.)
- Problem 4:
Both games can be stated in terms of repeated success/failure
trials.
Solving the problem in this manner (by interpreting it as
a success/failure problem) is quite easy.
- Problem 5:
The key here is to rephrase, in terms of S/F sequences, the
events described in (a) and (b).
- Problem 6:
The problem asks for a conditional probability; using the definition of a
conditional probability, this boils down to
a fairly standard success/failure probability computation.
- Problem 7:
Consider separately the cases when the series is over in exactly 4,
exactly 5, and exactly 6 games. First
ask yourself what it means that the series is over in exactly 4 games
and compute the probability for that event within the S/F
framework (using "assembly/level" computations).
Then do the same for the event "series is over in exactly 5 games"
and "series is over in exactly 6 games".
(There exists an alternative, slicker approach that requires only a
single computation. See if you find it, but no hints for this one!
The systematic method described above is perfectly acceptable.)
- Problem 8:
This is a two stage problem.
The first stage requires setting up an explicit sample space Omega, and a
set A, consisting of certain tuples;
the second stage is an easy S/F exercise.
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Last modified Mon 10 Feb 2003 05:31:52 PM CST