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**Theoretical Probability**

Quick Quiz Theoretical Probability

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**Find the theoretical probability of rolling a 2 or 3.**

Question #1 Find the theoretical probability of rolling a 2 or 3.

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Question #2 A bag contains 36 red, 48 green, 22 yellow, and 19 purple blocks. You pick one block from the bag at random. Find the theoretical probability. P(purple)

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Question #3 A bag contains 36 red, 48 green, 22 yellow, and 19 purple blocks. You pick one block from the bag at random. Find the theoretical probability. P(green or yellow)

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HW p.42 #4 If you guess the answer at random, what is the probability of getting at least three correct answers on a five-question true-or-false quiz?

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HW p.43 #18 The dart lands in a green region.

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HW p.43 #22 A caterer knows that, on average, there will be one broken egg in every 3 cartons. A carton contains 12 eggs. The caterer plans to serve 1200 eggs at a breakfast. What is the best estimate for the number of cartons the caterer should buy? 97 cartons 100 cartons 103 cartons 112 cartons

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HW p.44 #40 Team A has won one game and team B has won three games in a World Series. What is the experimental probability that team A wins the next game? That team B wins the next game? Do you think that experimental probability is a good predictor of the winner of the next game? Explain.

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**6.7: Permutations & Combinations**

GOAL: Be able to apply permutations and combinations.

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BIGELOW’S PIZZERIA Welcome to Bigelow’s Pizzeria!! We offer 4 types of pizza and 8 different pizza toppings. As our star customer, you get to choose the toppings below. Pizza Types Pizza Toppings Cheese White Thin Crust Stuffed Crust

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BIGELOW’S PIZZERIA Wait, seriously?? You want an 8-slice cheese pizza with a different topping on each slice? Write your toppings into each slice the way you want them arranged: If the positioning of the toppings on the slices doesn’t matter, how many ways can you choose 8 toppings out of the 8 toppings? If the positioning of the toppings on the slices does matter, how many ways can you arrange the 8 toppings on the slices?

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Permutations! A permutation is an arrangement of items in a particular order!

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Permutations! n! means n factorial, multiplying by every positive integer from 1 to n

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Example 1! Trophies! In how many ways can you arrange six trophies on a shelf?!

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Example 2! Dogs! In how many orders can ten dogs line up to be groomed?!

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Permutations! Above, we have been focusing on the order of every object given! We won’t always! The number of permutations of n items of a set arranged r times at a time is nPr 𝑛 𝑃 𝑟 = 𝑛! 𝑛−𝑟 ! for 0≤𝑟≤𝑛

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Example 3! Boat Race! Seven boats enter a race! First, second, and third places will be given to the three fastest boats! How many arrangements of first, second, and third places are possible with seven boats?! With ten boats?!

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Example 4! Pizza Slices Your pizza’s ready! You want to share your custom pizza with 3 friends! The four of you each take one slice! How many possibilities are there for the toppings that all four of you end up eating?! What if there were five of you?!

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Combinations A selection r of n total objects in which order does not matter is a combination 𝑛 𝐶 𝑟 = 𝑛! 𝑟! 𝑛−𝑟 ! for 0≤𝑟≤𝑛

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Example 5 Evaluate the expression. 12 𝐶 3

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Examples 6 - 7 Evaluate the expressions. 6. 8 𝐶 8 7. 9 𝐶 1

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**Example 8 Book Combinations**

A reading list for English class has 20 books on it. In how many ways can you choose four books to read?

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**Example 9 Voting for Class President**

Ten students are running for class president. You may vote for as many as three candidates. In how many ways can you vote for three or fewer candidates?

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Example 10 More Pizza! You want a thin crust pizza with 3 different toppings. How many different combinations of toppings are possible for your pizza?

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Example 11 You want 4 different toppings on any type of pizza. How many different pizza combinations are possible?

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TOTD What are the differences between combinations and permutations? (write a paragraph)

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TOTD Permutations There are 41 cars parked in the parking lot. How many ways could you pick a favorite, second favorite, third favorite, and fourth favorite car?

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Expected value a weighted average of all possible values where the weights are the probabilities of each outcome :

Expected value a weighted average of all possible values where the weights are the probabilities of each outcome :

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