Warm Up In your own words define probability. If you toss a coin 10 times, how many times SHOULD if come up heads? How did you prepare for your test yesterday? How do you think you did?

The Fundamental Counting Principle

Probability: The likelihood that an event will happen Compare the chance that a specific event will happen to all the possible events that could happen

Some Terms! Theoretical probability vs. experimental probability What should happen vs. what actually happens Outcome: the result of an experiment Sample Space: all the possible outcomes of an experiment Trial: one iteration of an experiment

Take out a coin! Heads Tails Total 10 Flips Total 1 Flip per student Probability 10 Flips Total 1 Flip per student 10 Flips per student

Law of Large Numbers As more trials are included in an experiment, the experimental probability of a particular outcome more closely approaches the theoretical probability of that outcome.

Fundamental Counting Principle Suppose that two events occur in order The first can occur in m ways The second can occur in n ways Then the two events can occur in m x n ways Keep doing this for all the events that occur # of ways the first event can occur # of ways the second event can occur # of ways the third event can occur

Multiply those bad boys. Example 1 A fast food restaurant sells hot dogs, hamburgers, chicken sandwiches, and barbecue sandwiches. They offer as sides french fries, hushpuppies, or onion rings. How many possible combinations are there? Multiply those bad boys. __________________ Event 1: Choose an entrée. __________________ Event 2: Choose a side.

Example 2 A mechanic offers three types oil changes (standard, synthetic, and high mileage), two types of wiper blades (low profile, and heavy use) and two types of mufflers (chrome and matte black). How many possible combinations are there?

Example 3 An ice cream store offers three types of cones and 31 flavors. How many different single-cone ice-cream cones is it possible to buy at this store?

Example 4: A little more challenging! In a certain state, automobile license plates display three letters followed by three digits. How many such plates are possible if repetition of the letter is allowed?

Solution 26 10 By the Fundamental Counting Principle, the number of possible license plates is… 26 x 26 x 26 x 10 x 10 x 10 17, 576, 000

Example 5: Stepping it up! In a certain state, automobile license plates display three letters followed by three digits. How many such plates are possible if repetition of the letters is not allowed?

Solution 26 25 24 10 By the Fundamental Counting Principle, the number of possible license plates is… 26 x 25 x 24 x 10 x 10 x 10 15, 600, 000

Factorial Notation! Product of n and each previous natural number (integers greater than 0). Denoted by n! and is called “n factorial” 3! = 3 x 2 x 1 8! = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1

Example 6 In how many different ways can a race with six runners be completed? Assume there is no tie.

Solution Six possible choices for first place Five possible choices for second place Four choices for third place and so on… So, by the Fundamental Counting Principle the number of different ways the race can be completed is… 6 x 5 x 4 x 3 x 2 x 1 = 6! = 720