Chapter 7 Sets

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Chapter 7Sets Probability

• Section 7.5
• Conditional Probability
• Independent Events

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• Lie detectors are not admitted as evidence in
  courtroom trials due to the fact they are not
  100 reliable. An experiment was conducted in
  which a group of suspects was instructed to lie
  or tell the truth to a set of questions, and a
  group of polygraph experts, along with the
  polygraph (lie detector), judged whether the
  suspect was telling the truth or not. The
  results are tabulated below.

Suspects Answers
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• Estimate the probability that
• a.) there will be a miscarriage of justice.
• b.) a suspect is a liar and gets away with the
  lie.
• c.) a suspect who is telling the truth is found
  to be honest
• by the experts.

Suspects Answers
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Conditional Probability

• A probability problem in which the sample space
  is reduced by known, or given, information is
  called a conditional probability. In other
  words, a conditional probability exists when the
  sample space has been limited to only those
  outcomes that fulfill a certain condition.

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• In a newspaper poll concerning violence on
  television, 600 people were asked, What is your
  opinion of the amount of violence on prime-time
  television is there too much violence on
  television? Their responses are indicated in
  the table below.

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Too Much Violence on Television?
Use the table to find the probabilities below. P
(Y) P (M) P (Y M) P (M Y) P (Y ? M) P
(M ? Y)
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A pair of dice is rolled. Find the probabilities
of the given events. a.) The sum is 12 b.)
The sum is 12, given that the sum is even c.)
The sum is 12, given that the sum is odd
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A pair of dice is rolled. Find the probabilities
of the given events. d.) The sum is even, given
that the sum is 12 e.) The sum is 4, given
that the sum is less than 6 f.) The sum is
less than 6, given that the sum is 4
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Product Rule of Probability
The Product Rule gives a method for finding the
probability that events E and F both occur.
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Example

• Two cards are drawn without replacement from a
  standard deck of 52 cards.
• a.) Find the probability of getting a King
• followed by an Ace.

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Example

• Two cards are drawn without replacement from a
  standard deck of 52 cards.
• b.) Find the probability of drawing a 7 and
• a Jack.

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Example

• Two cards are drawn without replacement from a
  standard deck of 52 cards.
• c.) Find the probability of drawing two Aces.

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• The Nissota Automobile Company buys emergency
  flashers from two different manufacturers one
  in Arkansas and one in Nevada.
• Thirty-nine percent of its turn-signal indicators
  are
• purchased from the Arkansas manufacturer, and
  the
• rest are purchased from the Nevada
  manufacturer.
• Two percent of the Arkansas turn-signal
  indicators are
• defective, and 1.7 of the Nevada indicators are
  
• defective.
• a.) What percent of the indicators are defective
  and
• made in Arkansas?
• b.) What percent of the indicators are
  defective?
• c.) What percent of the defective indicators are
  made
• in Nevada?

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• The Nissota Automobile Company buys emergency
  flashers from two different manufacturers one
  in Arkansas and one in Nevada.
• Thirty-nine percent of its turn-signal indicators
  are purchased from the Arkansas
• manufacturer, and the rest are purchased from
  the Nevada manufacturer.
• Two percent of the Arkansas turn-signal
  indicators are defective, and 1.7 of the
• Nevada indicators are defective.

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a.) What percent of the indicators are defective
and made in Arkansas? b.) What percent of the
indicators are defective? c.) What percent of
the defective indicators are made in Nevada?
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Two cards are dealt from a full deck of 52. Find
the probabilities of the given events. (Hint
Make a tree diagram, labeling each branch with
the appropriate probabilities.) a.) The first
card is a king. b.) Both cards are
kings. c.) The second card is a king, given
that the first card was a king. d.)
The second card is a king
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A coin is flipped twice in succession. Find the
probabilities of the given events. a.) Both
tosses result in tails b.) The second toss is
tails given the first toss is tails c.) The
second toss results in tails d.) Getting
tails on one toss and heads on the other
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Independent Events

• Two events are independent if the probability of
  one event does not depend / affect the
  probability (or occurrence) of the other event.

In other words, knowing F does not affect Es
probability.
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Product Rule for Independent Events
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Dependent Events

• Two events are dependent if the probability of
  one event does affect the probability (or
  likelihood of occurrence) of the other event.
• Two events E and F are dependent if
• P(E F) ? P(E) or P(F E) ? P(F)

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• Use your own personal experience or probabilities
  to determine whether the following events E and F
  are mutually exclusive and/or independent.
• a.) E is the event being a doctor and F is the
  event being
• a woman.
• b.) E is the event its raining and F is the
  event its
• sunny.
• c.) E is the event being single and F is the
  event being
• married.
• d.) E is the event having naturally blond hair
  and F is the
• event having naturally black hair.
• e.) If a die is rolled once, and E is the event
  getting a 4
• and F is the event getting an odd number.
• f.) If a die is rolled once, and E is the event
  getting a 4
• and F is the event getting an even number.

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A skateboard manufacturer buys 23 of its ball
bearing from a supplier in Akron, 38 from one in
Atlanta, and the rest from a supplier in Los
Angeles. Of the ball bearings from Akron, 4 are
defective 6.5 of those from Atlanta are
defective and 8.1 of those from Los Angeles are
defective. a.) Find the probability that a ball
bearing is defective. b.) Are the events
defective and from the Los Angeles
supplier independent? Show mathematical
justification. c.) Are the events defective
and from the Atlanta supplier
independent? Show mathematical justification.
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• A skateboard manufacturer buys 23 of its ball
  bearing from a supplier in Akron, 38 from one in
  Atlanta, and the rest from a supplier in Los
  Angeles. Of the ball bearings from Akron, 4 are
  defective 6.5 of those from Atlanta are
  defective and 8.1 of those from Los Angeles are
  defective.

a.) Find the probability that a ball bearing is
defective.
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b.) Are the events defective and from the Los
Angeles supplier independent? Show
mathematical justification. c.) Are the
events defective and from the Atlanta
supplier independent? Show mathematical
justification.