Probability of Independent and Dependent Events

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Probability of Independent and Dependent Events

• p. 730

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• Two events are Independent if the occurrence of 1
  has no effect on the occurrence of the other. (a
  coin toss 2 times, the first toss has no effect
  on the 2nd toss)

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Probability of Two Independent Events(can be
extended to probability of 3 or more ind. Events)

• A B are independent events then the probability
  that both A B occur is
• P(A and B) P(A) P(B)

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Probability of 2 Independent events

• You spin a wheel like the one on p. 730.
• During your turn you get to spin the wheel twice.
  What is the probability that you get more than
  500 on your first spin and then go bankrupt on
  your second spin?
• A spin gt 500 on 1st
• B bankrupt on 2nd
• P(A and B) P(A) P(B) 8/24 2/24
• 1/36 0.028

Both are ind. events
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BASEBALL

• During the 1997 baseball season, the Florida
  Marlins won 5 out of 7 home games and 3 out of 7
  away games against the San Francisco Giants.
  During the 1997 National League Division Series
  with the Giants, the Marlins played the first two
  games at home and the third game away. The
  Marlins won all three games.
• Estimate the probability of this happening. _
  Source The Florida Marlins

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• Let A, B, C be winning the 1st, 2nd, 3rd
  games
• The three events are independent and have
  experimental probabilities based on the regular
  season games.
• P(ABC) P(A)P(B)P(C)
• 5/7 5/7 3/7 75/343
• .219

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Using a Complement to Find a Probability

• You collect hockey trading cards. For one team
  there are 25 different cards in the set, and you
  have all of them except for the starting goalie
  card. To try and get this card, you buy 8 packs
  of 5 cards each. All cards in a pack are
  different and each of the cards is equally likely
  to be in a given pack.
• Find the probability that you will get at least
  one starting goalie card.

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• In one pack the probability of not getting the
  starting goalie card is
• P(no starting goalie)
• Buying packs of cards are independent events, so
  the probability of getting at least one starting
  goalie card in the 8 packs is
• P(at least one starting goalie) 1 -
  P(no starting goalie in any pack)8
• .832

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PROBABILITIES OF DEPENDENT EVENTS

• Two events A and B are dependent events if the
  occurrence of one affects the occurrence of the
  other.
• The probability that B will occur given that A
  has occurred is called the conditional
  probability of B given A and is written P(BA).

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Probability of Dependent Events

• If A B are dependant events, then the
  probability that both A B occur is
• P(AB) P(A) P(B/A)

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Finding Conditional Probabilities

• The table shows the number of endangered and
  threatened animal species in the United States as
  of November 30, 1998.
• Find (a) the probability that a listed animal is
  a reptile and (b) the probability that an
  endangered animal is a reptile.
• _ Source United States Fish and Wildlife Service

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• (a) P(reptile) number of reptiles 35
  total number of animals 475
• .0737
• (b) P(reptile/endangeres)
• Number of endangered reptiles 14
  total num endangered animals
  322
• .0394

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Comparing Dependent and Independent Events

• You randomly select two cards from a standard
  52-card deck. What is the probability that the
  first card is not a face card (a king, queen, or
  jack) and the second card is a face card if
• (a) you replace the first card before selecting
  the second, and
• (b) you do not replace the first card?

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• (A) If you replace the first card before
  selecting the second card, then A and B are
  independent events. So, the probability is
• P(A and B) P(A) P(B) 40 12 30
  52 52 169
• 0.178
• (B) If you do not replace the first card before
  selecting the second card, then A and B are
  dependent events. So, the probability is
• P(A and B) P(A) P(BA) 4012 40
  52 51 221
• .0181

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Probability of Three Dependent Events

• You and two friends go to a restaurant and order
  a sandwich. The menu has 10 types of sandwiches
  and each of you is equally likely to order any
  type. What is the probability that each of you
  orders a different type?

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• Let event A be that you order a sandwich, event B
  be that one friend orders a different type, and
  event C be that your other friend orders a third
  type. These events are dependent. So, the
  probability that each of you orders a different
  type is
• P(A and B and C)
• P(A) P(BA) P(CA and B)
• 10/10 9/10 8/10
• 18/25 .72

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Assignment