Chapter 7 Sets

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Chapter 7 Sets Probability

• Section 7.4
• Basic Concepts of Probability

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• To determine the probability of the union of two
  events E and F in a sample space S, we must use
  the Union Rule for Probability, which is based on
  the Union Rule for Sets.
• Union Rule for Probability
• For any events E and F from a sample space S,
• P(E ? F) P(E) P(F) P(E ? F)

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• Example If a single card is randomly drawn from
  an ordinary deck of 52 cards, find the
  probability that it will be a spade or a face
  card.
• P(E ? F) P(E) P(F) P(E ? F)
• P(spade ? face) P(spade) P(face) P(spade ?
  face)
• P(spade ? face) 13 12 - 3 22
  11 .
• 52 52 52 52 26

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Possible Outcomes When Two Fair Dice Are Thrown
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• Example Two fair dice are thrown. Find the
  probability that the sum of the dice is a number
  greater than 9 or that the first die is a 5.
• P(E ? F) P(E) P(F) P(E ? F)
• P(sumgt9 ? 1st 5) P(sumgt9) P(1st5)
  P(sumgt9 ? 1st5)
• P(sumgt9 ? 1st 5) 6 6 - 2
  10 5 .
• 36 36 36 36 18

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• If events E and F are mutually exclusive, then E
  ? F ? by defintion
• therefore, P(E ? F) 0.
• Union Rule for Mutually Exclusive Events
• For mutually exclusive events E and F,
• P(E ? F) P(E) P(F)

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• An event E and its complement E? are mutually
  exclusive.
• E ? E? ?
• The union of an event E and its complement E? are
  equal to the sample space.
• E ? E? S
• Complement Rule
• P(E) P(E ?) 1

8

• Example Find the probability that a card drawn
  from a standard deck will be larger than a 3.
  (Aces are high.)
• Let E card is larger than a 3
• Let E? card is a 3 or less (card is a 2 or 3)
• Find P(E? ).
• P(E? ) 4 4 8 2 .
• 52 52
  13
• Using the Complement Rule, P(E) P(E ?)
  1
• P(E) 2 1
• 13
• THUS, P(E) 11 .
• 13

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Odds

• Sometimes probability statements are given in
  terms of odds, a comparison of P(E) P(E ?) .
• ODDS
• The odds in favor of an event E are defined as
  the ratio of P(E) to P(E ?), where P(E ?) ? 0.
• P(E)
• P(E ?)

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• Example Suppose the sports analysts say the
  probability that Alabama will win the NCAA
  National Championship this year is 3/5. Find the
  odds in favor of UA becoming National Champs this
  year.
• P(E) 3/5 3 ,
• P(E ?) 2/5 2
• which is written 3 to 2 or 32 .
• Note ALWAYS write odds using to or

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Converting Odds to Probability

• If the odds favoring event E are m to n, then
• P(E) m and P(E ?) n .
• m n
  m n

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• Example If the odds in favor of Smarty Jones
  winning the Triple Crown next year are 4 to 5,
  what is the probability that he will win the
  Triple Crown?
• Triple Crown Kentucky Derby, the Preakness,
  and Belmont
• The odds indicate chances 4 out of 9 ( 4 5)
  that he will win, so
• P(winning) 4 .
• 9
• Note Racetrack odds are generally against a
  horse winning.

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Empirical vs. Theoretical Probability

• In many real-life problems, it is not possible to
  establish exact probabilities for events.
  Instead, useful approximations are found by using
  experimentation or past experiences. This kind
  of probability is known as empirical probability.
• Theoretical probability is the true probability
  of an event, while empirical probability is the
  probability of the event that had been determined
  through trials or observances.

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• Example Charlie Brown knows that the
  probability of getting tails when he tosses an
  unbiased coin is 0.5, but when he actually tosses
  the coin 10 times, he finds his probability for
  getting tails is actually 0.3.
• Which probability represents the empirical
  probability of the event tails?
• 0.3

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Properties of Probability

• In a sample space, the probability of each
  individual outcome is a number between 0 and 1.
• 0 ? P(E) ? 1
• The sum of all the probabilities in a sample
  space is always equal to 1.