Study Guide for Exam 2
Finite Math, Spring 2012
To prepare for the second exam, you should review chapter 6 in the textbook. You should
also review your homework problems, as many exam problems will resemble homework you’ve
done. As a general guide, I recommend reviewing the following topics.
Chapter 6: Probability
• Definitions
– An experiment is the thing we do, which produces an outcome.
– Every time we actually do the experiment is called a trial.
– The set of all outcomes is the sample space.
– An event is a subset of the sample space.
– The impossible event is ∅, which has probability 0.
– If the sample space is S, then the certainty event is S itself, which has probability 1.
– Two events E and F are disjoint or mutually exclusive if they cannot both
happen, i.e. if E ∩ F = ∅.
• Be able to translate events described in words into set notation (A ∩ B, A ∪ B, A0 ,
etc.) and vice versa.
• Compute empirical or experimental probability: if you do the experiment N times
and event E occurs f (E) times, then the experimental/empirical probability is f (E)/N .
• Important facts about probability:
– For every event E, it is true that 0 ≤ p(E) ≤ 1.
– If the probabilities of the outcomes are p1 , p2 , . . . , pn , then p1 + p2 + · · · + pn = 1.
– If E and F are events, then p(E ∪ F ) = p(E) + p(F ) − p(E ∩ F ).
• Use Venn diagrams to reason about probability.
Exam 2 Study Guide
Page 2 of 3
Finite Math, Spring 2012
• Calculating probabilities
– If your experiment has equally likely outcomes, then p(E) =
p(good) =
n(E)
; in other words,
n(S)
# good ways
.
total # ways
– Use your counting skills from Chapter 5 to do such problems.
– Remember, the words “at least” or “at most” mean you will have to break the
problem into different options and add. For example, “at most two” means “zero
or one or two.”
– The Complement Rule often helps:
p(good) = 1 − p(bad).
• Conditional Probability
– The probability of event E if we already know that event F happened is called the
conditional probability. It is written “p(E|F )” and pronounced “the probability of E given F .” The way to compute it is
p(E|F ) =
p(E ∩ F )
.
p(F )
– Remember that p(E|F ) 6= p(F |E).
– Two events E and F might or might not be independent.
∗ Intuitively, E and F are independent if one does not affect the other, whereas
they are not independent if one influences the other somehow.
∗ Mathematically, E and F are independent if p(E|F ) = p(E).
∗ Saying “E and F are independent” also means p(F |E) = p(F ).
∗ Most importantly, E and F are independent when p(E ∩ F ) = p(E)p(F ).
• Warnings
– BE CAREFUL! It is not true that p(E ∩ F ) = p(E)p(F ), unless the events are
independent.
– BE CAREFUL! It is not true that p(E ∪ F ) = p(E) + p(F ); rather, p(E ∪ F ) =
p(E) + p(F ) − p(E ∩ F ).
– BE CAREFUL! Independent events and mutually exclusive events are not
the same thing.
Events E and F are mutually exclusive if p(E ∩ F ) = 0;
they are independent if p(E ∩ F ) = p(E)p(F ).
Exam 2 Study Guide
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Finite Math, Spring 2012
• Tree Diagrams and Bayes’s Theorem
– Be able to draw the tree diagram corresponding to a situation.
∗ Each branch of the tree gets a probability.
∗ Then you multiply along all the whole path to get the probability of each
final result.
– Remember that p(E|F ) and p(F |E) are not the same thing.
– (Bayes’s Theorem) If you know p(E|F ) and you want to find p(F |E),
1. Draw the tree diagram. (F will be to the left of E.)
2. Calculate all the probabilities.
p(E ∩ F )
.
3. Compute p(F |E) =
p(E)
(To find p(E), you will have to add up several branches of the tree.)