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## Counting Methods and Probability

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**Counting Methods and Probability**Chapter10.1-10.3**10.1 Counting Principles and Permutations**• Determine how many different possibilities are possible: • 1. There are 3 different ice cream flavors and 5 different toppings. You can have one type of ice cream and one topping. • 2. You have 30 different shirts, 8 types of pants, and 4 different types of shoes. How many different ways can you dress yourself? • 3. You have just enough money to go out to eat and see a movie. There are 5 different restaurants near the movie theater and 10 different movies playing.**Fundamental Counting Principle**• If three events occur in m, n, and p ways, then the number of ways that all three events can occur is m x n x p. • It can be ANY number of events.**Determine how many different possibilities are possible if:**• a. repetition is allowed • b. repetition is not allowed. • 1. A 4-digit lock with numbers 0-9. • 2. A 6-digit lottery with numbers from 1-30. • 3. A license plate with 3 letters followed by 4 numbers.**Permutations**• How many ways can you pick r things out of n, where ORDER MATTERS. • nPr= • You can plug these into your calculator. MATHPRBnPr.**Find the number of Permutations:**• First, use the Fundamental Counting Principle. • Then, use the Permutations Formula by hand. • 1. A TV news director has 8 news stories to present on the evening news. • a. How many different ways can the stories be presented? • b. If only 3 stories will be presented, how many possible ways can a lead story, a second story, and a closing story be presented?**Find the number of Permutations:**• First, use the Fundamental Counting Principle. • Then, use the Permutations Formula by hand. • 2. 10 students at Norwin are running for President. • a. How many different ways can the students give their speeches to the school? • b. First place becomes President, second place becomes Vice-President, third place becomes Treasurer, and fourth place becomes Secretary. How many ways can the students be P, VP, T, and S?**Permutations with Repetition:**• How many different permutations can you make with the following letters: • 1. ABCD • 2. ABCC • 3. ABBB**Permutations with Repetition:**• different permutations of n objects where one object is repeated s1 times, another repeated s2 times, and so on is:**Find the number of different permutations of the letters in:**• 1. KAYAK • 2. TALLAHASSEE • 3. CINCINNATI**Homework**• 10.1 #11-16, 32-53x3, 64-66**10.2 Combinations**• Place 44, 50, and 64-66 on the board. Show your work! • At your seats, answer the following questions: • 1. How many different ways can Ms. Rothrauff call on students to write the above answers on the board? • 2. How many different ways can you pick 4 lunch sides given that there are 10 options? • Standard Deck has 4 different suits (hearts, diamonds, spades, clubs) and 13 cards of each suit (2-10, Jack, Queen, King, and Ace).**Permutations with Repetition:**• How many different permutations can you make with the following letters: • 1. ABCD • 2. ABCC • 3. ABBB • How does this help prove this is true?**0!=1?**• The definition of a factorial is n!=n x (n-1)! • Use this information to prove that 0!=1.**Finding the Formula for Combinations**• Discuss in your groups what you think the formula will be for Combinations (where order DOES NOT MATTER). • Consider the following: • Permutation Formula from yesterday. • Different Permutations of ABCD picking all 4 letters. • If order DID NOT MATTER, how many different possibilities would there be to order ABCD using all 4 letters?**Combinations**• How many ways can you pick r things out of n, where order DOES NOT MATTER. • nCr= • You can plug these into your calculator. MATHPRBnCr.**Multiple Events**• When finding the number of ways both event A AND event B can occur, you need to multiply. • When finding the number of ways that event A OR event B can occur, you add instead. • Pg 691**Subtracting Possibilities**• Counting problems that involve phrases like “at least” or “at most” are sometimes easier to solve by subtracting possibilities you do not want from the total number of possibilities. • Pg 691**Multiple Events Example:**• The Norwin Student Senate consists of 6 seniors, 5 juniors, 4 sophomores, and 3 freshman. • a. How many different committees of exactly 2 seniors and 2 juniors can be chosen? • b. How many different committees of at most 4 students can be chosen?**Subtracting Possibilities Example:**• You are going to toss 10 different coins. How many different ways will at least 4 of the coins show heads?**Finding card combinations:**• In a standard deck of 52 cards: • 1. How many ways can you get a flush in hearts? • 2. How many ways can you get all red cards? • Standard Deck has 4 different suits (hearts, diamonds, spades, clubs) and 13 cards of each suit (2-10, Jack, Queen, King, and Ace). • *flush=all same suit (hearts, diamonds, etc.)**Finding card combinations (cont):**• 3. How many ways can you get at most one heart? • 4. How many ways can you get at least one 6? • Standard Deck has 4 different suits (hearts, diamonds, spades, clubs) and 13 cards of each suit (2-10, Jack, Queen, King, and Ace).**Homework**• 10.2 #3-10, 13-18**10.2 Binomial Theorem**• Place numbers 14, 16, and 18 on the board. Show your work! • At your seats, answer the following question: • How many ways can you get a full house with a standard deck of 52 cards? • Standard Deck has 4 different suits (hearts, diamonds, spades, clubs) and 13 cards of each suit (2-10, Jack, Queen, King, and Ace). • *full house=3 of the same type and 2 of the same type (QQQKK, 444JJ, 33399, etc.)**Pascal’s Triangle**• Refer to page 692 in your books. • If you arrange the values of nCrin triangular pattern in which each row corresponds to a value of n, you get Pascal’s Triangle. • The r corresponds to the number in that row. • *You start counting with 0. Both the rows and the number in that row.* • *0C0 = 1 and is the 0th row.***Use Pascal’s Triangle:**• 1. From a collection of 7 baseball caps, you want to trade 3. Use Pascal’s Triangle to find the number of combinations of 3 caps that can be traded. • 2. The 7 members of the math club chose 2 members to represent them at a meeting. Use Pascal’s triangle to find the number of combinations of 2 members that can be chosen as representatives.**Binomial Theorem**• Refer to page 693 in your books. • Steps to use the Binomial Theorem: • 1. Identify a, b, and n. • 2. Make a list of all the C terms vertically. n=n for all C terms, while r starts at 0 at the top and goes to n on the bottom. (There will be n+1 C terms.) • 3. Next to each C term, write the a term in parenthesis. Raise each a term starting at the top to the nth power down to the bottom ending with 0th power. • 4. Next to the a term, write the b term in parenthesis. Raise each b term starting at the top to the 0th power down to the bottom ending with the nth power. • 5. Multiply all of the terms out and put a “+” between each new term.**BT Examples:**• 1. (x+y)6 • 2. (5-2y)3 • 3. (3x-2)4**Find a Coefficient in an Expansion:**• Find the coefficient of xr in the expansion of (a+b)n. • Formula: nCrarbn-r**Coefficient Examples:**• 1. Find the coefficient of x5 in the expansion of (x-3)7. • 2. Find the coefficient of y3 in the expansion of (5+2y)8. • 3. Find the coefficient of x3y4 in the expansion of (2x-y)7.**Homework**• 10.2 #19-33odd, 38-39, 48-49**BT Review**• Place 25 and 31 on the board. Show your work!!! • At your seats, try 24 and 26 on page 695.**10.3 Probability and Odds**• 1. How many different possibilities are there to win a lottery if 3 numbers are drawn from 1-15… • a. With repetition? • b. Without repetition? • 2. What would be the probability of winning the lottery… • a. With repetition? • b. Without repetition?**Probability**• Theoretical Probability of event A: • P(A)= • Experimental Probability of event A: • P(A)=**Card Probabilities:**• You pick a card from a standard deck of 52 cards. Find the following probabilities: • 1. Picking an heart. • 2. Picking a red King. • 3. Picking anything but an Ace. • 4. Picking a number card (2-9). • 5. Picking a Joker. • Standard Deck has 4 different suits (hearts, diamonds, spades, clubs) and 13 cards of each suit (2-10, Jack, Queen, King, and Ace).**Number Probabilities:**• You have an equally likely chance of picking any integer from 1-20. Find the probabilities: • 1. Picking a perfect square. • 2. Picking a factor of 30. • 3. Picking a multiple of 3.**Odds**• Odds in favor of event A= • Odds against event A=**Card Odds:**• You pick a card from a standard deck of 52 cards. Find the following odds: • 1. Odds in favor of drawing a 5. • 2. Odds against drawing a diamond. • 3. Odds in favor of drawing a heart. • 4. Odds against drawing a Queen. • Standard Deck has 4 different suits (hearts, diamonds, spades, clubs) and 13 cards of each suit (2-10, Jack, Queen, King, and Ace).**Geometric Probability**• You throw a dart at the board. Your dart is equally likely to hit any point inside of the board. What is the probability of getting 0 points? What is the probability of getting 50 points? Are you more likely to get 0 points or 50 points? 3in 3in3in 50pts 0pts**Homework**• 10.3 #4-18even, 20-23, 35-39