Chapter 8 Counting Principles: Further Probability Topics

1
Chapter 8Counting Principles Further
Probability Topics

• Section 8.3
• Probability Applications
• of Counting Principles

2

• Recall that tree diagrams can be used to solve
  probability problems involving dependent events,
  but tree diagrams sometimes require a large
  number of branches.
• Permutations and combinations are especially
  helpful when the numbers involved are large.

3

• The use of combinations to solve probability
  problems depends on the basic probability
  principle
• Let S be a sample space with equally likely
  outcomes, and let event E be a subset of S. Then
  the probability that event E occurs, written
  P(E), is
• where n(E) and n(S) represent the number of
  elements in sets E and S.

4

• Combinations tend to be used more often than
  permutations in probability because many times
  the final result does not depend on the order of
  events.
• However, order does matter in some cases, and
  permutations would need to be used.
• In some cases, a mixture of permutations and
  combinations are needed to solve the problem.

5

• Example You and your friends order twelve
  burritos to go from a Mexican restaurant, five
  with hot peppers and seven without however, the
  restaurant forgot to label them. If you pick
  three burritos at random, what is the probability
  that all three have hot peppers?
• Find the number of ways you can choose 3
  burritos from 12.
• 12C3 220 This is your sample
  space.
• Find the number of ways you can choose 3
  burritos with hot peppers.
• 5C3 7C0 10
• P(3 w/hot peppers) __10__ 0.045
• 220

6

• Example A basketball coach was asked to rank
  order his top three players from the starting
  five players. Suppose one of the players is
  named Jack.
• a.) What is the probability that Jack is one of
  the top
• three players?
• Find the number of ways the top 3 players can be
  chosen.
• 5C3 10 This is your sample space.
• Find the number of ways that Jack could be
  chosen as one of the three ranked players.
• 4C2 6 Since we want Jack to be one of the
  three chosen players, we only need to find
  out how many ways the coach can select the other
  two from the remaining 4 players.
• Determine the probability Jack is one of the top
  three.
• P(top three) 6 0.6
• 10

7

• Example A basketball coach was asked to rank
  order his top three players from the starting
  five players. Suppose one of the players is
  named Jack.
• b.) What is the probability that Jack is ranked
  1?
• Find the number of ways the top 3 players can be
  ranked.
• 5P3 60 This is your sample space.
• Find the number of ways that Jack could be chosen
  as the
• number one ranked player.
• _1_ 4 3 12
• Since we want Jack to be the number one ranked
  player, we only need to find
• out how many ways the coach can select the second
  and third ranked players
• from the remaining 4.
• Determine the probability Jack is the top-ranked
  player.
• P( rank 1 ) 12 0.2
• 60

8

• Example A standard poker hand consists of 5
  cards. Find the probability that a person is
  dealt a straight flush (5 in a row in a single
  suit, but not a royal flush). Assume Aces are
  either high or low.
• A flush could start with an Ace, 2, 3, 4, 5, 6,
  7, 8, or 9.
• This gives 9 choices in each of 4 suits, so
  there are 36 choices in all. ( 9 4 36 )
• P(straight flush)

___36____ 52 C 5
36______ 2, 598, 960

-5
0.00001385 1.385 x 10
9

• Example Two hundred people apply for three
  jobs. Sixty of the applicants are women. If
  three persons are selected at random, what is the
  probability that
• a.) all are women?
• b.) exactly two are men?
• c.) at least one is female?