Thinking Mathematically

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Thinking Mathematically

• 11.4 Fundamentals of Probability

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• If you toss a coin 3 times, what are the chances
  of getting heads exactly 2 times?
• We answer the above using probability. We need
  to define some terms
• In conducting an experiment (like tossing a
  coin) there are
• outcomes
• sample space set of all possible outcomes
  
• event a particular set of outcomes

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• Tossing a coin

Outcome/ Sample Space
Event 2 Heads
3rd Toss
2nd Toss
1st Toss
HHH
HHT
HTH
HTT
THH
THT
TTH
TTT
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Computing Theoretical Probability
If all outcomes in an experiment are equally
likely, then the probability of an event E is
computed as follows

Previous example E event of getting two
heads P(E) 3/8
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• Example A bag contains 9 balls 2 red, 3 blue
  and 4 black. Suppose a ball is drawn at random.
  Whats the probability that it is
• Red?
• Blue?
• Not blue?

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Example Computing Theoretical Probability

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

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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Empirical Probability
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• Example The table shows the breakdown of the 89
  thousand single parents on active duty in the
  U.S. military in 2002. All numbers are in
  thousands and rounded to the nearest thousand.
  Find the probability that a randomly selected
  single parent in the U.S. military is a male.

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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

HW page 585 1, 5, 9, 13, 17, 21, 25, 29, 37,
41, 45, 47, 49, 59, 61