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of Law total probabilities
Archie is taking a course in probability
theory. Discussion sessions with the
class TA (teaching assistant) conflict
with Archie's trips to the beach. Archie
has to decide between attending TA
sessions, which might help him do better
in the probability course, and going to
the beach. Archie learns that
% of
past students who did not attend TA
sessions received a grade of B or below
in the course. Archie also learns that
% of past students received a grade
higher than B. After some thought,
Archie decides to attend the TA
sessions. Not all of his classmates share
his decision; in fact, only
out of every
of his classmates attend discussion
sessions with the TA. Compute the
probability that Archie will receive a
grade higher than a B in the course.
Round your answer to two decimal
places.
(If necessary, consult a list of formulas.)
We'll examine the information in the above paragraph.
We'll write
for the event "achieves a grade higher than a B." We
are told that the probability of this event is
. We'll write
discussion sessions with the TA," and we'll write
We are told that
out of every
for the event "meets in
for the complementary event.
students attends discussion sessions with the TA;
hence
, so that
. We are also given information as to the
lack of success of students who do not meet with the TA. In particular, we are given that
,
so
.
Applied to this scenario, the Law of total probability reads
.
Filling in all the numerical information, we see that
.
Solving, we obtain
.
For additional explanation,
see:
Essentials
of Business
Statistics,
3rd Ed.
Bowerman
et al.
Section 4.5: Bayes' Theorem (Optional)
Thus, the answer is
.
Law of total probabilities
Archie is stranded in a war zone in the
land of the Xhosas. His only chance of
survival is to reach Sudland, which lies
miles away. There are just two
routes to Sudland; one is through a
desert, and the other is a long and
perilous trek through the mountains.
Both routes are terribly dangerous. In
the past,
% of all who tried to reach
Sudland failed. Of all attempts to reach
Sudland,
out of every
were by the
desert route. Although it is the longer
route, the mountains seem to provide
more success:
% of those who
attempted to reach Sudland through the
mountains were successful. To save
time, Archie intends to go via the desert.
What are his chances making of it
through to Sudland? Round your answer
to two decimal places.
(If necessary, consult a list of formulas.)
It is necessary to decode the information given.
Let us write
for the event "successfully reaches Sudland from Xhosa." We
are given that the probability of this event is
. There are just two
routes: through the mountains or through the desert. Only
out of every
attempts are made by trekking through the desert; that is,
Consequently,
.
. Given the choice of the mountain route, the
probability of reaching Sudland is given as
The law of total probability applied to the current situation reads
.
.
Filling in the numerical information, we see that
.
We deduce that
.
For additional explanation,
see:
Essentials
of Business
Statistics,
3rd Ed.
Bowerman
et al.
Section 4.5: Bayes' Theorem (Optional)
Thus, the answer is
.
Law of total probabilities
Archie must travel to a foreign country
where a deadly strain of flu has ravaged
the population. Already,
% of the
population has succumbed to the flu and
died. There is a vaccine, but because
the flu is extremely contagious, it
spreads quite rapidly, and to date only
out of every
people have been
vaccinated. Thirty percent of those who
are not vaccinated die from the flu.
Archie has just enough time to be
vaccinated before he leaves. What is the
probability that Archie will succumb to
the flu and die while he is away? Round
your answer to two decimal places.
(If necessary, consult a list of formulas.)
The inhabitants of the foreign country can be partitioned into two non-overlapping
classes, namely those who have been vaccinated and those who have not. We
represent the event "has been vaccinated" by the abbreviation
, and we
represent the event "has not been vaccinated" by the abbreviation
are told that the event
occurs with probability
. The conditional
probability that a non-vaccinated person will die from the flu is given as
is,
. We
; that
. We shall also need to use the fact that
out of every
people have been vaccinated, which we translate as
.
The law of total probability applied to the current situation reads
.
Filling in the numerical information, we see that
.
We deduce that
.
For additional explanation,
see:
Essentials
of Business
Statistics,
3rd Ed.
Bowerman
et al.
Section 4.5: Bayes' Theorem (Optional)
Thus, the answer is
.

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CSU Sacramento - ACCY - 171

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