Excursions in Modern Mathematics Sixth Edition

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Excursions in Modern MathematicsSixth Edition

• Peter Tannenbaum

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Chapter 15Chances, Probabilities, and Odds

• Measuring Uncertainty

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Chances, Probabilities, and OddsOutline/learning
Objectives

• To describe an appropriate sample space of a
  random experiment.
• To apply the multiplication rule, permutations,
  and combinations to counting problems.
• To understand the concept of a probability
  assignment.

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Chances, Probabilities, and OddsOutline/learning
Objectives

• To identify independent events and their
  properties.
• To use the language of odds in describing
  probabilities of events.

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Chances, Probabilities, and Odds

• 15.1 Random Experiments and Sample Spaces

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Chances, Probabilities, and Odds

• Random experiment
• Description of an activity or process whose
  outcome cannot be predicted ahead of time.
• Sample space
• Associated with every random experiment is the
  set of all of its possible outcomes. We will
  consistently use the letter S to denote a sample
  space and N to denote its size (the number of
  outcomes in S).

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Chances, Probabilities, and Odds
Rolling the Dice Part 1 One of the most common
things we do with dice is to roll a pair of dice
and consider just the total of the the two die.
A more general scenario is when we do care what
number each individual turns up. Here below we
have a sample space with 36 different outcomes.
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Chances, Probabilities, and Odds
Rolling the Dice Part 1 When looking at the
figure below you will notice that we are treating
the dice as distinguishable objects (as if one
were white and the other red), so that
and are considered different
outcomes.
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Chances, Probabilities, and Odds

• 15.2 Counting Sample Spaces

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Chances, Probabilities, and Odds

• The Multiplication Rule
• When something is done in stages, the number of
  ways it can be done is found by multiplying the
  number of ways each of the stages can be done.

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Chances, Probabilities, and Odds

• The Making of a Wardrobe Part 2
• Our strategy will be to think of an outfit as
  being put together in stages and to draw a box
  for each of the stages. We then separately count
  the number of choices at each stage and enter
  that number in the corresponding box.

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Chances, Probabilities, and Odds

• The Making of a Wardrobe Part 2
• The last step is to multiply the numbers in each
  box. The final count for the number of different
  outfits is
• N 3 ? 7 ? 27 ? 3 1701

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Chances, Probabilities, and Odds

• 15.3 Permutations and Combinations

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Chances, Probabilities, and Odds

• Permutation
• A group of objects where the ordering of the
  objects within the group makes a difference.
• Combination
• A group of objects in which the ordering of the
  objects is irrelevant.

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Chances, Probabilities, and Odds

• The Pleasures of Ice Cream Part 1
• Say you want a true double in a bowl how many
  different choices so you have?

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Chances, Probabilities, and Odds

• The Pleasures of Ice Cream Part 1
• The natural impulse is to count the number of
  choices using the multiplication rule (and a box
  model) as shown below. This would give an answer
  of 930.

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Chances, Probabilities, and Odds

• The Pleasures of Ice Cream Part 1
• Unfortunately, this answer is double counting
  each of the true doubles. Why?

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Chances, Probabilities, and Odds

• The Pleasures of Ice Cream Part 1
• When we use the multiplication rule, there is a
  well-defined order to things, and a scoop of
  strawberry followed by a scoop of chocolate is
  counted separate from a scoop of chocolate
  followed by a scoop of strawberry.
• The good news is that now we understand why the
  count of 930 is wrong and we can fix it. All we
  have to do is divide the original count by 2.
• (31 ? 30)/2 465

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Chances, Probabilities, and Odds

• 15.4 Probability Spaces

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Chances, Probabilities, and Odds

• Event
• Any subset of the sample space.
• Simple event
• An event that consists of just one outcome.
• Impossible event
• A special case of the empty set ,
  corresponding to an event with no outcomes.

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Chances, Probabilities, and Odds

• Probability assignment
• A function that assigns to each event E a number
  between 0 and 1, which represents the probability
  of the event E and which we denote by Pr (E).
• Probability space
• Once a specific probability assignment is made on
  a sample space, the combination of the sample
  space and the probability assignment.

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Chances, Probabilities, and Odds

• Elements of a Probability Space
• Sample space S o1, o2,., oN
• Probability assignment Pr(o1),Pr(o2), Pr(oN)
• Each of these is a number between 0 and 1
  satisfying Pr(o1) Pr(o2) Pr(oN) 1
• Events These are all the subsets of S,
  including and S itself. The probability of
  an event is given by the sum of the probabilities
  of the individual outcomes that make up the
  event. In particular, Pr( ) 0 and
  Pr(S) 1

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Chances, Probabilities, and Odds

• 15.5 Equiprobable Spaces

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Chances, Probabilities, and Odds

• Probabilities in Equiprobable Spaces
• Pr(E) k/N (where k denotes the size of the
  event E and N denotes the size of the sample
  space S).
• A probability space where each simple event has
  an equal probability is called an equiprobable
  equal opportunity space.

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Chances, Probabilities, and Odds

• Rolling the Dice Part 2
• The sample space has N 36 individual outcomes,
  each with probability 1/36. We will use the
  notation T2, T3, T12 to describe the events
  roll a total of 2, roll a total of 3, ,
  roll a total of 12, respectively. We show you
  how to find Pr(T7) and Pr(T11),
• T11 , Thus,
• Pr(T11) 2/36 ? 0.056
• T7
  , Thus,
• Pr(T7) 6/36 1/6 ? 0.167

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Chances, Probabilities, and Odds

• Tallying
• We can just write down all the individual
  outcomes in the event E and tally their number.
  This approach gives
• and Pr(E) 11/36.

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Chances, Probabilities, and Odds

• Complementary Event
• Imagine that you are playing a game, and you win
  if at least one of the two numbers comes up an
  Ace (thats event E). Otherwise you lose (call
  that event F). The two events E and F are called
  complementary events. The probabilities of
  complementary events add up to 1. Thus,
• Pr(E) 1 Pr(F).

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Chances, Probabilities, and Odds

• Independence Events
• If the occurrence of one event does not affect
  the probability of the occurrence of the the
  other.
• Multiplication Principle for Independent Events
• When events E and F are independent, the
  probability that both occur is the product of
  their respective probabilities in other words,
• Pr (E and F) Pr(E) Pr(F).

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Chances, Probabilities, and Odds

• 15.6 Odds

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Chances, Probabilities, and Odds

• Odds
• Let E be an arbitrary event. If F denotes the
  number of ways that event E can occur (the
  favorable outcomes or hits), and U denotes the
  number of ways that event E does not occur (the
  unfavorable outcomes, or misses), then the odds
  of (also called the odds in favor of), the event
  E are given by the ratio F to U, and the odds
  against the event E are given by the ratio U to
  F.

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Chances, Probabilities, and Odds
Conclusion

• Sample space
• Random experiment
• Events
• Probability assignment
• Equiprobable spaces