Math 110.107, Calculus II (Biological and Social Sciences)
Fall 2010 Course Lecture Synopses
Week 12: November 15 through November 19
http://www.mathematics.jhu.edu/brown/courses/f10/107.htm
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MWF 10:00am - 10:50am Krieger 205 |
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403 Krieger Hall |
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410-516-8179 |
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Office Hours:
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M |
1:00-2:00 pm |
by appt. other times |
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W |
1:00-2:00 pm |
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Below is some basic information pertaining to the lectures of this course. I will update this page after each lecture or two to benefit both the students in their attempts to organize the material for the course, and the TAs so that they know what material I covered and how it was covered. Please direct any comments about this page to me at the above contact information. |
· Monday,
November 15: Here, I continued the discussion of chapter 12 but
venturing into Section 12.2. I started
with a definition of a random experiment,
and a sample space, and an event, again using examples for motivation. We played with common set notation to explore
how to create complicated events (subsets of the sample space) from unions,
intersections and complements of simply events.
I then reviewed De Morgan’s Laws.
I discussed when two events are disjoint,
(and when many events are pairwise
disjoint), and defined what is a probability. Really, given any finite set (the sample
space), a probability is really only as functional assignment
that satisfies
three properties: (1) for any event
,
(its image
under the probability function is in the unit interval), (2)
, and
, and (3) for any two disjoint events,
and
,
. Really, any
assignment that satisfies these tenets is a probability. I gave some examples, and discussed two
consequences: (1) For any event
,
, and (2) For any two (not necessarily disjoint)
events
and
,
. The only thing to remember for the last one is that
two simply add the probabilities, one would be including the outcomes in both
events twice. The last term is a way to
fix that. Assuming that any of the
outcomes in a sample space
are equally likely to occur, one can than
easily calculate the probability of any outcome, and then any event:
, where
is the number
of outcomes in the event
. I then
ventured into the notion of Conditional Probability, a way to calculate the
probability of an event occurring when it is already known that another event
has occurred. In essence, when an event
occurs, how
does that change the probability that a second event can occur. When they are related, it can change a
lot. I talked about how this works, and
developed the equation to calculate the conditional probability that event
occurs given
that event
has
occurred:
. Essentially,
if event
has occurred,
then the sample space is now limited to
. The only
outcomes left in
are the ones
that are also in
. That is the
top of the fraction. The bottom is to
address the idea that the space of available outcomes is not only
. For equally
likely outcomes in
, we get
.
· Wednesday, November 17:
· Friday, November 19: