Transcript Expected Value

Chapter
9
Probability
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9-4 Odds, Conditional Probability, and
Expected Value
 Computing Odds
 Conditional Probabilities
 Expected Value
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Computing Odds
Let P(A) be the probability that A occurs and P(A)
be the probability that A does not occur. Then the
odds in favor of an event A are
and the odds against an event A are
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Computing Odds
In the case of equally like outcomes,
odds in favor
odds against
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Example 9-12
Find the odds in favor of the event occurring:
a. Rolling a number less than 5 on a die
4 : 2 or 2 : 1
b. Tossing heads on a fair coin
1:1
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Example 9-12 cont
Find the odds in favor of the event occurring:
c. Drawing an ace from an ordinary 52-card deck
4 : 48 or 1 : 12
d. Drawing a heart from an ordinary 52-card deck
13 : 39 or 1 : 3
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Example 9-13
Find the probability of
making totally black
copies if the odds are 3
to 1 against making
totally black copies.
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Computing Odds
If the odds in favor of event E are m : n, then
If the odds against event E are m : n, then
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Conditional Probabilities
If A and B are events in sample space S and
P(A)  0, then the conditional probability that an
event B occurs given that event A has occurred is
given by
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Example 9-14
What is the probability of rolling a 6 on a fair die if
you know that the roll is an even number?
If event B is rolling a 6 and event A is rolling an
even number, then
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Expected Value
If, in an experiment, the possible outcomes are
numbers
occurring with probabilities
respectively, then the expected value
(mathematical expectation) E is given by
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Expected Value
 Expected value can be used to predict the
average result of an experiment when it is
repeated many times.
 Expected value cannot be used to determine the
outcome of any single experiment.
Fair game:
When payoffs are involved and the net winnings
are $0 (the expected value minus cost to play a
game of chance), the game is a fair game.
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Example 9-15
Suppose you pay $5.00 to play the following
game. Two coins are tossed. You receive $10 if
two heads occur, $5 if exactly one head occurs,
and nothing if no heads appear. Is this a fair
game? That is, are the net winnings $0?
The net winnings are $0, so this is a fair game.
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