12'4 Probability of Compound Events

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12.4 Probability of Compound Events

• P. 724

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Mutually Exclusive Events
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Intersection of A B
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• To find P(A or B) you must consider what
  outcomes, if any, are in the intersection of A
  and B.
• If there are none, then A and B are mutually
  exclusive events and P(A or B) P(A)P(B)
• If A and B are not mutually exclusive, then the
  outcomes in the intersection or A B are counted
  twice when P(A) P(B) are added.
• So P(A B) must be subtracted once from the sum

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EXAMPLE 1

• One six-sided die is rolled.
• What is the probability of rolling a multiple of
  3 or 5?
• P(A or B) P(A) P(B) 2/6 1/6 1/2
• 0.5

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EXAMPLE 2

• One six-sided die is rolled. What is the
  probability of rolling a multiple of 3 or a
  multiple of 2?
• A Mult 3 2 outcomes
• B mult 2 3 outcomes
• P(A or B) P(A) P(B) P(AB)
• P(A or B) 2/6 3/6 1/6
• 2/3 0.67

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EXAMPLE 3

• In a poll of high school juniors, 6 out of 15
  took a French class and 11 out of 15 took a math
  class.
• Fourteen out of 15 students took French or math.
• What is the probability that a student took both
  French and math?

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• A French
• B Math
• P(A) 6/15, P(B) 11/15, P(AorB) 14/15
• P(A or B) P(A) P(B) P(AB)
• 14/15 6/15 11/15 P(A B)
• P(A B) 6/15 11/15 14/15
• P(A B) 3/15 1/5 .20

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Using complements to find Probability

• The event A, called the complement of event A,
  consists of all outcomes that are not in A.
• The notation A is read A prime.

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Probability of the complement of an event

• The probability of the complement of A is
• P(A) 1 - P(A)

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EXAMPLE 4

• A card is randomly selected from a standard deck
  of 52 cards.
• Find the probability of the given event.
• a. The card is not a king.
• P(K) 4/52 so P(K)
• 48/52 0.923

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• b. The card is not an ace or a jack.
• P(not ace or Jack) 1-(P(4/52 4/52))
• 1- 8/52
• 44/52 0.846

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• In a survey of 200 pet owners, 103 owned dogs, 88
  owned cats, 25 owned birds, and 18 owned
  reptiles.
• 1. None of the respondents owned both a cat and a
  bird.
• What is the probability that they owned a cat or
  a bird?
• 113/200
• 0.565
• 2. Of the respondents, 52 owned both a cat and a
  dog.
• What is the probability that a respondent owned a
  cat or a dog?
• 139/200
• 0.695

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• 3. Of the respondents, 119 owned a dog or a
  reptile.
• What is the probability that they owned a dog and
  a reptile?
• 1/100 0.010

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Assignment