Counting Principles and Probability We equate probability with

Counting Principles and Probability

We equate probability with “what are my chances…. ”

Counting Principles and Probability We must learn to “count” before we can calculate probability!

Your familiar with the situation: You have 4 Counting Principles tops, 5 hats and 6 pairs of pants. Assuming How many outcomes are possible? everything matches how many different outfits How many unique outcomes are possible? can you make? This is a CHAIN of events each How many outcomes that make sense are possible? with its own number of possible outcomes. So the total number of possible outcomes would be 4· 5· 6 which is equal to 120 outcomes.

FUNDAMENTAL COUNTING PRINCIPLE If you have a chain of evens that each have m outcomes the possible outcomes of the entire “chain” would be the product of the individual events that make up this chain.

Counting Principles How many possible ways are there to get a sum of 7?

How many possible ways are there to get a sum of 7?

Counting Principles How many ways can we draw a card from the bag containing numbers 1 -14 REPLACE it, then draw again AND get those two numbers to sum to 17? 7 6 14 13 14 3 12 11 10 2 12 ways 9 13 4 5 12 5 11 10 6 7 1 8 9 8

How many ways can we draw a card from the bag containing numbers 1 -14 WITHOUT REPLACEMENT, then draw again AND get those two numbers to sum to 17? 7 6 5 4 3 2 1 14 13 12 11 10 9 8 14 3 13 4 6 ways 11 10 12 5 6 7 9 8

FOR EXAMPLE: How many 3 element lottery tickets can be created if the first element must be a number from 15, the second must be a letter from A-G and the third must be a number from 10 -12? 105 ways 5· 7· 3 5 A B 4 3 2 C D E 10 5 11 A 1 F 12 3 G

FOR EXAMPLE: How many 3 element lottery tickets can be created if the numbers must be picked are 1 - 14 in any order, no repeats? 14 13 12 2184 ways 7 6 5 4 3 2 1 14 13 12 11 10 9 8

FOR EXAMPLE: How many 3 element lottery tickets can be created if the numbers must be picked are 1 - 14 in any order repeats okay? 14 14 14 2744 ways 7 6 5 4 3 2 1 14 13 12 11 10 9 8

What if you are counting outcomes and order matters? Like in a race (first, second, third? ) When ORDER MATTERS it is called PERMUTATION.

How many different finishes can there be for the Sausages? 5 4 3

How many different finishes can there be for the Sausages? This could also be determined using the formula P n r

How many different finishes can there be for the Sausages? This could also be determined using the formula P 5 3

How many different finishes can there be for the Sausages? This could also be determined using the formula P 5 3 60

BANANA How many different combinations of the word BANANA can you make?

BANANA 6 But there would be repeats in there because you couldn’t tell one A 5 from 4 another 3 A. 2 So if 1 we want DISTINGUISHABLE 720(DISTINGUISHABLE outcomes PERMUTATION) we’d have to get rid of repeats. 6!

BANANA 60

Remember: When order matters (when it’s more important to be first than second…) it’s permutation.

If order doesn’t matter it’s a combination! Picking volunteers from a larger group Picking committee members from a larger group

Permutation or Combination? Picking prom court C Sitting in open seating at a concert Lining up for school lunch P Choosing the president and vp of junior class P C C Poker hand. 12 person swim team from a team of 30 C 6 person relay team P

If order doesn’t matter it’s a combination! C n r

A standard poker hand consists of 5 cards dealt from a deck of 52. How many different poker hands are possible. C 52 5 2, 598, 960

You are forming a 12 member team from 10 girls and 15 boys. The team must have five girls and seven boys. How many different 12 member teams are possible? · C C 10 5 15 7 1, 620