Thinking Mathematically

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Thinking Mathematically

• Fundamentals of Probability

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Computing Theoretical Probability

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Example of Theoretical Probability

• What is the probability of getting at most 2
  heads when a coin is tossed 3 times?

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Solution

• The probability of getting at most 2 heads when a
  coin is tossed 3 times is 7/8

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Example Computing Theoretical Probability

• A die is rolled once. Find the Probability of
  getting a number less than 5.

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Solution

• The event of getting a number less than 5 can
  occur in 4 ways 1, 2, 3, 4.
• P(less than 5)
• (number of ways a number less than 5 can occur)
  (total number of possible outcomes)
• 4/6 2/3

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Example Probability and a Deck of 52 Cards

• You are dealt one card from a standard 52-card
  deck. Find the probability of being dealt a
  King.

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Solution

• Because there are 52 cards, the total number of
  possible ways of being dealt a single card is 52.
  We use 52, the total number of possible outcomes,
  as the number in the denominator. Because there
  are 4 kings in the deck, the event of being dealt
  a king can occur 4 ways.
• P(king) 4/52 1/13

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Example Probabilities in Genetics

• Each person carries two genes that are related
  to the absence or presence of the disease cystic
  fibrosis. 1 in 25 Americans carries one normal
  gene and one defective gene. If we use c to
  represent a defective gene and C a normal gene,
  such a carrier can be designated as Cc. Thus, CC
  is a person who neither carries nor has cystic
  fibrosis, Cc is a carrier who is not actually
  sick, and cc is a person sick with the disease.
  One copy of each gene is passed on to the child
  from the parents. If each parent carries one
  cystic fibrosis gene, what is the probability
  that their child will have cystic fibrosis?

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Solution

• The table shows the four equally likely outcomes.
  Only the cc child will develop cystic fibrosis.
• Thus, P(cystic fibrosis) 1/4
• If each parent carries one cystic fibrosis gene,
  the probability that their child will have cystic
  fibrosis is 1/4.

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Empirical Probability
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Example Computing Empirical Probability

• There are approximately 3 million Arab Americans
  in America. The circle graph shows that the
  majority of Arab Americans are Christians. If an
  Arab American is selected at random, find the
  empirical probability of selecting a Catholic.

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Solution

• The probability of selecting a Catholic is the
  observed number of Arab Americans who are
  Catholic, 1.26 (million), divided by the total
  number of Arab Americans, 3 (million).
• P(selecting a Catholic from the Arab American
  Population) 1.26/3 0.42

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Thinking Mathematically

• Fundamentals of Probability