Write

1
Write

• I need 3 people to help move desks around, so we
  put everyones name into a hat and randomly pick
  3.
• We need to choose a president, vice-president,
  and secretary for the class, so we put everyones
  name into a hat and randomly pick president, then
  vice-president, then secretary.
• In what ways are these situations the same? How
  are they different?

2
Counting Outcomes

• Wednesday 28 November 2007

3
The Sample Space for an Event

• Identify the sample space if I roll two dice,
  add, and then flip a coin.

4
Sample Space Tree Diagram
5
Theoretical Probability

• A probability is a number between 0 and 1 that
  describes the likelihood that an event will take
  place.

6
Factorials
7
Order

• Sometimes, the order that we choose things
  matters.
• President
• Vice President
• Secretary
• Adam then Liz then Corey
• Liz then Corey then Adam

8
Order

• Sometimes, the order that we choose things
  doesnt matter.
• Three people to help move desks.
• Adam then Liz then Corey
• Liz then Corey then Adam

9
Permutation

• Order matters!
• 6321 is different from 1362
• How many arrangements of r things can I make
  from a set of n?

10
Permutation

• How many different ways might the Gold, Silver,
  and Bronze medals in the Mr. Math USA
  competition be handed out?

11
Permutation

• The number of different arrangements of r things
  taken from a set of n is equal to

12
Permutation

• How many different ways might the Gold, Silver,
  and Bronze medals in the Mr. Math USA
  competition be handed out?

13
Permutation Try Some

• How many ways can we choose 5 people from the
  class and line them up?
• If everyone runs for the door when the bell
  rings, how many different 1st-2nd-3rd possible
  finishes are there?
• If there are 265 seniors, how many different
  valedictorian-salutatorian combinations are
  there?
• How many different 6-letter license plates are
  there, if you cant repeat any letters?

14
Combination

• Order doesnt matter.
• Its not like a lock.
• How many different ways can I choose r things
  from a group of n?

15
Combinations

• How many ways are there to choose 4 states?
• How many ways are there to choose 4 states if I
  dont want any that border the Pacific?

16
Combination

• The number of different groups of r things that
  can be chosen from a set of n is equal to

17
Combinations

• How many ways are there to choose 4 states?
• How many ways are there to choose 4 states if I
  dont want any that border the Pacific?

18
Combination Try Some

• I need to choose 2 of the 87 Team 2 sophomores to
  be Honored Students for November. How many
  possible pairs are there?
• If we choose 6 of the numbers from 1 to 38 in a
  school lottery, how many possible winning
  combinations are there?
• How many different possible groups of 8 CDs can I
  choose from a group of 154?
• If there are 8 books on a shelf, how many
  different ways can I shove 3 index cards between
  them?

19
Permutations, Combinations, and Probability

• Since probability is about outcomes, successes,
  and failures
• And since permutations combinations help us
  count these things really quickly
• Used properly, p c can be a real time-saver in
  computing probabilities!

20
Combinations and Probability

• If I choose 4 states at random, what is the
  probability that none of them border on the
  Pacific ocean?

21
Combinations and Probability

• If there are 42 males and 47 females on Team 2,
  what is the probability that both of the Honored
  Students will be female?
• If we choose 2 teachers from Team 2, what is the
  probability that they are both male?
• What is the probability that, choosing 6 numbers
  from 1 to 38 in a lottery, none of them will be
  odd?

22
Summary

• Permutations count arrangements.
• Combinations count groups. (Order doesnt
  matter).
• They can help us to determine probabilities by
  counting outcomes.
• Homework p. 148, 18-28