Math 6.01-6.10

6.01 Random Lesson Stuff _________________________________________________________________________________________ Slide 1 ⬇ Jenna is having a party for her friends, and she is planning to play lots of games. There will be card games, board games, and even darts! In this lesson, you will follow Jenna as she calculates the probability of different events in each type of game. _________________________________________________________________________________________ Slide 2 ⬇ The first game at the party will use a number cube for players to move around the game board. When a player rolls a number cube numbered one through six, the possible outcomes are 1, 2, 3, 4, 5, and 6. Jenna assumes that the number cube is fair and that all numbers have an equal chance of being rolled. What are the chances that a player rolls a three? There is only one side of the number cube labeled "three" and there are a total of six possible outcomes. The chances of rolling a three are one out of six or 1/6. Another way to say this is the probability of rolling a three is 1/6. Outcomes: The possible results. Probability: A number from 0 to 1 that represents the likelihood of an event. Probability = number of favorable outcomesnumber of total possible outcomes What is the Probabillty of rolling an Odd Number on the six-sided number cube? There are three odd numbers on the number cube: 1, 3, and 5. These are the three favorable outcomes. There are six total possible outcomes when rolling a number cube. number of favorable outcomes number of total possible outcomes=3/6 The probability of rolling an odd number is 3/6, which simplifies to 1/2. Probability is a number from 0 to 1 and can be expressed as a fraction, a decimal, or a percent. For example, the probability of rolling an odd number on the number cube is 1/2, which is the same as saying 0.5 or 50%. Comparing Probabilities To compare probabilities, be sure that they are all in the same form. For example, when comparing the probability of 12 with a probability of 75%, try changing the 12 to a percent: 12 = 0.5 = 50% Now compare 50% to 75%, and you can see that a probability of 75% is greater than 50%. What is the probability that if a coin is tossed, it lands heads up? There are two possible outcomes when flipping a coin: heads and tails. Only one outcome is "heads," so the probability of landing heads up is number of favorable outcomesnumber of total possible outcomes = 12 = 0.5 = 50% A bag contains 2 red marbles and 3 white marbles of equal size. What is the probability of pulling a white marble out of the bag without looking? There are five possible outcomes when pulling a marble from the bag: red, red, white, white, white. Three outcomes are white, so the probability of pulling a white marble from the bag is number of favorable outcomesnumber of total possible outcomes = 35 = 0.6 = 60% The spinner has four equal sections. What is the probability of the spinner landing on the red space? There are four possible outcomes when spinning the spinner: red, yellow, green, and blue. Only one outcome is red, so the probability of landing on the red space is number of favorable outcomesnumber of total possible outcomes = 14 = 0.25 = 25% _________________________________________________________________________________________ Slide 3 ⬇ Jenna is planning to play card games with her friends, and she knows that probability can be used to describe the likelihood of something happening. Check out this video to see how probability is related to the likelihood of an event. There was a lot of vocabulary in there! The big idea is that the smaller the number, the less likely the event is to happen, and the larger the number, the more likely the event is to happen. _________________________________________________________________________________________ Slide 5 ⬇ Vocabulary Outcomes: The possible results. Probability: A number from 0 to 1 that represents the likelihood of an event. Probability = number of favorable outcomesnumber of total possible outcomes Summary Probability is a number from 0 to 1 and can be expressed as a fraction, a decimal, or a percent. Example: Two out of every five party guests is a boy. The probability of selecting a boy's name at random is number of favorable outcomesnumber of total possible outcomes =25 = 0.4 = 40% Events can be classified from impossible to certain. The smaller the number, the less likely the event. The larger the number, the more likely the event.

6.01 Questions From Lesson ______________________________________________________________________

6.02 Random Lesson Stuff _________________________________________________________________________________________ Slide 1 ⬇ Jenna has taught her dog to shake hands. When she says, "Shake," her dog Scoundrel will offer a paw. As guests arrived at her party, Jenna noticed that Scoundrel didn't always offer the same paw, so one day she decided to keep track of which paw Scoundrel offered to shake with. Jenna wants to predict how many times Scoundrel will offer the left paw if she shakes hands 100 times. How can Jenna make this prediction? _________________________________________________________________________________________ Slide 2 ⬇ Scoundrel used her left paw to shake hands 15 out of 24 times. Scoundrel is equally likely to offer either paw when asked to shake hands. The theoretical probability of Scoundrel offering the left paw is 12 = 0.5 = 50%. The data Jenna collected is not the same as the theoretical probability. But, she can use her data to calculate the experimental probability of Scoundrel offering the left paw. Theoretical probability: number of favorable outcomesnumber of total possible outcomes Experimental probability: Number of times an outcome occursTotal number of times the experiment is completed The experimental probability is equal to number of times left paw is offeredtotal number of trials = 1524 = 0.625 = 62.5%. Another term for experimental probability is relative frequency. _________________________________________________________________________________________ Slide 3 ⬇ Jenna is trying to predict how many times Scoundrel will offer her left paw in 100 trials. How will her predictions be different if she uses the theoretical probability compared to the experimental probability? Theoretical probability There is an equally likely chance that Scoundrel will offer either paw when asked. The theoretical probability of Scoundrel offering her left paw is 50%. 50%•100=0.50•100=50 times. After 100 trials, Jenna can expect that Scoundrel will offer the left paw 50 times. Experimental probability During the party, Scoundrel offered her left paw 62.5% of the time. 62.5%•100=0.625•100=62.5 times. It wouldn't make sense to offer a paw 0.5 times, so round 62.5 to 63. After 100 trials, Jenna can expect that Scoundrel will offer the left paw 63 times. Compare the probabilities The experimental and theoretical probabilities are not equal. In Jenna's data, each group's experimental probability was very different from the theoretical probability. If Jenna collected data many more times, her experimental probability will get closer to the theoretical probability. This is because of the law of large numbers. The law of large numbers means that the more times an experiment is conducted, the closer the experimental probability will get to the theoretical probability. Theoretical probability is not always equal to experimental probability. _________________________________________________________________________________________ Slide 4 ⬇ All of this probablity reminded Jenna of an experiment her class did. There were four groups of students and each group had a bag of 20 marbles, but they didn't know the colors of the marbles. Each group picked out a marble, recorded the color, and put the marble back in the bag. They recorded their data for 50 trials in the table shown below: Mr. Alexander asked his class to predict how many blue marbles they would expect to draw if they did the experiment 1,000 times. How did they make this prediction? First, they needed to figure out the experimental probability of drawing a blue marble. Experimental probability is also known as relative frequency.Relative Frequency of the Blue Marbles What is the relative frequency of blue marbles for each group? To calculate the relative frequency or experimental probability of drawing a blue marble, divide the number of times a blue marble was drawn by the total number of trials for each group. _________________________________________________________________________________________ Slide 6 ⬇In Mr. Alexander's marble-drawing experiment, each group of students had a different experimental probability of drawing a blue marble. Then, when the class combined their data for 200 trials, they calculated a 45% chance of drawing a blue marble. Mr. Alexander asked the class to predict how many blue marbles they would draw after 1,000 trials. How many times would you expect a blue marble to be drawn after 1,000 trials? Select each card to see how each group might answer this question. At the end of the experiment, Mr. Alexander asked the students to take all of the marbles out of the bag and record the colors and their counts. Only then did the students learn that each bag contained 4 green marbles, 6 red marbles, and 10 blue marbles. What is the probability of drawing a blue marble from the bag based on this information? Probability of drawing a blue marble= number of blue marblestotal number of marbles Since there are 20 marbles in the bag and 10 marbles are blue, the probability of drawing a blue marble is 1020 =0.50, which is 50%. This is called the theoretical probability. Theoretical probability is the likeliness of an event happening based on all of the possible outcomes. _________________________________________________________________________________________ Slide 8 ⬇ Vocabulary Experimental probability: The number of times an outcome occurs divided by the total number of times the experiment is completed. Relative frequency: The number of times an outcome occurs divided by the total number of times the experiment is completed. Summary Theoretical probability is determined by the fraction number of favorable outcomesnumber of total possible outcomes . You can predict the number of times an event will occur by multiplying the theoretical probability by the total number of trials. Relative frequency and experimental probability both mean the same thing number of times an outcome occurstotal number of times the experiment is completed. This fraction can also be represented as a percentage by first changing the fraction to a decimal, and then multiplying the result by 100 to convert the decimal to a percent. The more trials that are conducted, the closer the experimental probability gets to the theoretical probability.

6.02 Questions From Lesson ______________________________________________________________________ Slide 2 ⬇ What is the experimental probability of Scoundrel offering the right paw? The experimental probability is equal to number of times right paw is offered total number of trials=9/24= 0.375 = 37.5%. ______________________________________________________________________

6.04 Random Lesson Stuff _________________________________________________________________________________________ Slide 1 ⬇ A popular game at Jenna's party is darts. Jenna's dartboard is a square with three circles on it. If players hit the board with a dart, they can earn 20, 25, or 50 points depending on where the dart lands. _________________________________________________________________________________________ Slide 2 ⬇ In Jenna's dartboard, the side of the square is 20 inches. The radius of the largest circle is 10 inches, the radius of the middle circle is 5 inches, and the radius of the smallest circle is 2.5 inches. If a player hits the board with a dart, what is the probability that it will land inside the small circle to earn 50 points? The best way to look at this is by using our knowledge of probability, together with finding area. Recall that probability is defined as number of favorable outcomesnumber of possible outcomes The total area of the dartboard will represent the number of possible outcomes. The dartboard is a square, so the area of the dartboard is A = s2 = (20 in)2 = 400 in2 The player will win 50 points if they hit the innermost circle. The area of the innermost circle will represent the number of favorable outcomes. The area of the innermost circle is A = πr2 = π(2.5 in)2 = π(6.25 in2) ≈ 3.14(6.25 in2) = 19.625 in2. The probability of hitting the innermost circle is number of favorable outcomesnumber of possible outcomes = 19.625 in2400 in2 ≈0.049, which is about 4.9%. The probability that a player will win 50 points by throwing a dart that lands inside the innermost circle is roughly 5%. _________________________________________________________________________________________ Slide 3 ⬇ Another game at the party is played with a spinner like the one shown. The spinner is supposed to be a fair spinner, with each sector having an equal probability. Jenna noticed that the spinner has landed on the green sector 10 times in a row. What is the probability that it lands on green on the next spin? The probability of the spinner landing on green on any spin is 14 , or 25%, because there is one green section out of four total sections. Even though the spinner landed on green the last 10 spins, the probability of landing on green on the next spin is not changed. Each spin is independent from the spin before. There are 25 guests at the party: 14 girls and 11 boys. The table shows what the guests chose to eat: Jenna randomly selected a name from the guest list.

6.04 Questions From Lesson ______________________________________________________________________ Slide 2⬇ What is the probability that a player throwing a dart earns at least 25 points by landing inside the middle circle? The radius of the 25-point circle is 5 inches. A = πr2Area formulaA = π(5 in)2Substitute the radius into the formula for the area of a circle.A = π (25 in2)Simplify.A ≈ 3.14(25 in2)Substitute 3.14 as an approximation for π.A = 78.5 in2Simplify. The area of the circle is 78.5 in2. Remember that the area of the dartboard is: A = s2 = (20 in)2 = 400 in2. So the probability of hitting the 25-point circle is number of favorable outcomesnumber of possible outcomes = 78.5 in2400 in2 ≈0.19625 The probability of getting at least 25 points is roughly 20%. What is the probability that a player throwing a dart earns at least 10 points by landing inside the largest circle? The radius of the 10-point circle is 10 inches. A = πr2Area formulaA = π(10 in)2Substitute the radius into the formula for the area of a circle.A = π (100 in2)Simplify.A ≈ 3.14(100 in2)Substitute 3.14 as an approximation for π.A = 314 in2Simplify. The area of the circle is 314 in2. Remember that the area of the dartboard is: A = s2 = (20 in)2 = 400 in2. So the probability of hitting the 10-point circle is number of favorable outcomesnumber of possible outcomes = 314 in2400 in2 ≈0.785 The probability of getting at least 10 points is roughly 79%. ______________________________________________________________________ Slide 3 ⬇ Jenna rolled a number cube numbered one through six. It landed on an even number 25 times. What is the probability that it lands on an even number on the next roll? There are six numbers on the number cube, so the total number of outcomes is six. There are three even numbers on the number cube: 2, 4, 6. The number of favorable outcomes is three. The probability of rolling an even number on the next roll is number of favorable outcomesnumber of possible outcomes = 36 = 0.50 = 50% Jenna flipped a coin, and it landed tails up 75 times. What is the probability the coin lands tails up on the next flip? There are two possible outcomes when flipping a coin: heads and tails. There is only one side of the coin marked tails. The number of favorable outcomes is one. The probability of flipping a coin and having it land tails up is number of favorable outcomesnumber of possible outcomes = 12 = 0.50 = 50% ______________________________________________________________________ Slide 4 ⬇ What is the Probability that she chose a girl? The probability that she chose a girl is equal to number of favorable outcomesnumber of possible outcomes = number of girlsnumber of guests = 1425 = 0.56 = 56% What is the probability that she chose name of someone who ate pizza ? The probability that she chose someone who ate pizza is equal to number of favorable outcomesnumber of possible outcomes = number of pizza eatersnumber of guests = 1225 = 0.48 = 48% What is the Probability that she chose the name of a girl who ate pizza? The probability that she chose a girl who ate pizza is equal to number of favorable outcomes number of possible outcomes= number of girls who ate pizza number of guests=8/25= 0.32 = 32% The table shows the guests at the party and their food choices. What is the probability that a boy's name is randomly selected? The probability that a boy's name is randomly selected is equal to number of favorable outcomes number of possible outcomes= number of boys number of guests=11/25= 0.44 = 44% The table shows the guests at the party and their food choices. What is the probability that the person who is selected chose a burger? The probability that the person ate a burger is equal to number of favorable outcomes number of possible outcomes= number of burger eaters number of guests=13/25= 0.52 = 52% The table shows the guests at the party and their food choices. What is the probability that the person chosen is a boy who ate a burger? The probability that she chose a boy who ate a burger is equal to number of favorable outcomes number of possible outcomes= number of boys who ate burgers number of guests=7/25= 0.28 = 28% ______________________________________________________________________ Slide 5 ⬇ Summary Probability is defined as number of favorable outcomesnumber of possible outcomes Areas of geometric figures can be used as outcomes. The probability of hitting the innermost circle is number of favorable outcomesnumber of possible outcomes = 19.625 in2400 in2 ≈ 0.049, which is about 4.9%. Sometimes in repeated events such as flipping a coin or spinning a spinner, the same outcome will happen many times in a row. This does not change the probability of the next outcome. For example, a fair coin might land heads-up 100 times. The probability that it will land heads-up on the next flip is still 12 .

6.05 Random Lesson Stuff _________________________________________________________________________________________ Slide 1 ⬇ Jenna and her friends are playing a board game that uses two number cubes labeled 1 through 6. In this game, players get to roll again if they roll a 1 on either number cube. Jenna recorded the results from the last 10 rolls in the data table. Jenna wonders, what is the probability that a player will roll a 1? In this lesson, you will learn how Jenna can observe data to develop a probability model. _________________________________________________________________________________________ Slide 2 ⬇ Take a look at Jenna's data table. Based on Jenna's data, the experimental probability of rolling a 1 is number of ones rolledtotal number of rolls = 210 = 0.2 = 20% This is because two rolls out of 10 resulted in a 1.

6.05 Questions From Lesson ______________________________________________________________________ Slide 2 ⬇ How Closely does this match the theoretical probability of rolling in a 1? Since there are 11 ways to roll a 1 and 36 total possible outcomes, the theoretical probability of rolling a 1 is number of favorable outcomesnumber of total possible outcomes = 1136 ≈ 0.306 = 30.6% Experimental vs. Theoretical Probability Jenna's experimental probability of rolling a 1 is only 20%, while the theoretical probability is about 31%. Why the difference? One reason might be that Jenna's probability is based on a very small number of results. She only observed 10 rolls. The more data she collects, the closer her experimental probability will be to the theoretical probability of 30.6%. ______________________________________________________________________ Slide 3 ⬇ Jenna flipped a coin 375 times. It landed tails up 300 times. What is the experimental probability of a flip resulting in tails up? The coin landed tails up 300 times out of 375 flips. number of times an outcome occurstotal number of times the experiment is completed = 300375 = 0.8 = 80% Jenna rolled a number cube several times. The table shows the number of times each face of the cube appeared on top. What is the probability of rolling an odd number on this number cube? The odd numbers are 1, 3, and 5. An odd number appeared 6 + 0 + 3 = 9 times out of 24 rolls. The probability of rolling an odd number is number of favorable outcomesnumber of total possible outcomes = 924 = 0.375 = 37.5% ______________________________________________________________________ Slide 4 ⬇ Summary Experimental probabilities can be used to develop a probability model. Example: The probability of rolling an odd number on this number cube is 924 = 0.375 = 37.5%. The probability of rolling an even number on this number cube is 1524 = 0.625 = 62.5%.

7.01 Random Lesson Stuff _________________________________________________________________________________________ Slide 1 ⬇ Benjamin and Donna are planning to spend the day with their friends at the beach. First they have to go pick up their friends, but there is only room for one person in the front passenger seat. To decide who will get to sit in the front seat, they will flip two coins. If the two coins show the same side, Benjamin wins, and if the two coins show different sides, Donna will win. Is this a fair way to decide who sits in the front seat? In this lesson you will follow Benjamin and Donna on their beach trip and learn how to use lists, tables, and diagrams to determine the probabilities of compound events. _________________________________________________________________________________________ Slide 2 ⬇ Who gets to sit in the front seat? They decide to flip two coins to see who gets to sit in the front. If the two coins show the same side (two heads or two tails), then Benjamin gets to sit in front seat. If the two coins show different sides (one head and one tail), then Donna will sit in the front seat. To see if this is a fair way to decide, look at the probability of each outcome in the sample space. One way to create a sample space of all outcomes from flipping two coins is to make a table. There are two possible outcomes for Benjamin's coin toss: heads and tails. There are two possible outcomes for Donna's coin toss: heads and tails. To set up the table, pick one set of outcomes for the rows and the other set of outcomes for the columns. In the table below, the outcomes for Benjamin's coin are used for the columns: heads and tails. The outcomes for Donna's coin are used for the rows: heads and tails. The cell with the (heads, heads) outcome represents Benjamin's coin landing heads up and Donna's coin landing heads up. The cell with the (tails, heads) outcome represents Benjamin's coin landing tails up and Donna's coin landing heads up. Neither Benjamin's nor Donna's coin toss is affected by the other person's results. These two coins tosses are independent events. Taken together, these two independent events make up the compound event of two coins being tossed. All of the events in the table make up the sample space: (heads, heads), (tails, heads), (heads, tails), (tails, tails). Sample space: All possible outcomes or results. Compound event: A combination of two or more simple events. Independent event: Events that are not affected by a previous outcome. _________________________________________________________________________________________ Slide 3 ⬇ That was a very simple table showing the sample space of two coins being tossed. Try something a little more challenging. The spinner shown to the right is spun two times. Make a table of the sample space. For the first spin, the possible outcomes are Red, Yellow, Green, or Blue. For the second spin, the possible outcomes are also Red, Yellow, Green, or Blue. The shaded portion of the table shows the possible results of the two spins. This is also known as the sample space. _________________________________________________________________________________________ Slide 4 ⬇ Donna won the coin toss, so she gets to sit in the front seat. They go to pick up their friends, Allen and Carly. What is the probability that Allen, Benjamin, and Carly sit in the back seat in alphabetical order? Sometimes it doesn't make sense to make a table to show the outcomes in the sample space. For this example, an organized list would be more appropriate. To make things a little simpler, use the letters A, B, and C in place of the names Allen, Benjamin, and Carly. To make an organized list, it makes sense to start with A, for Allen, because that is the first letter of the alphabet. Start making a list of the different ways A, B, and C can sit in the back seat. Two seating arrangements have Allen sitting on the left. A - B - CA - C - B Two seating arrangements have Allen sitting in the middle. B - A - CC - A - B Two seating arrangements have Allen sitting on the right. B - C - AC - B - A There are a total of six different seating arrangements for the back seat. What is the probability that they are in alphabetical order: Allen, Benjamin, Carly? There is one favorable outcome, A - B - C , out of a total of six possible outcomes in the sample space. The probability that they are in alphabetical order is equal to: Number of favorable outcomesNumber of total possible outcomes = 16 =0.16≈17%. There is a 17% chance that Allen, Benjamin, and Carly will sit in alphabetical order. _________________________________________________________________________________________ Slide 5 ⬇ Now Donna is in the front seat and Allen, Benjamin, and Carly are in the back seat in alphabetical order. They have arrived at the beach and decide to get a bite to eat at the burger bar. There are three burger choices: veggie, turkey, or beef. There are also three choices for cheese: American, Swiss, or cheddar. What are the different types of burgers that can be made using these choices? A tree diagram would be useful here. Make a tree diagram starting with the burger choices: It is called a tree diagram because the different choices form branches. Each branch is an outcome of the sample space. Starting with the first branch, the outcomes are: (veggie, American), (veggie, Swiss), (veggie, cheddar). The second branch has the following outcomes: (turkey, American), (turkey, Swiss), (turkey, cheddar). Finally, the third branch has the following outcomes: (beef, American), (beef, Swiss), and (beef, cheddar). All the outcomes can be placed together and listed out to show the entire sample space, like this: (veggie, American), (veggie, Swiss), (veggie, cheddar), (turkey, American), (turkey, Swiss), (turkey, cheddar), (beef, American), (beef, Swiss), and (beef, cheddar). Tree diagram: A diagram that shows all possible outcomes of an event. _________________________________________________________________________________________ Slide 7⬇ Vocabulary Sample space: All possible outcomes or results. Compound event: A combination of two or more simple events. Independent event: Events that are not affected by a previous outcome. Tree diagram: A diagram that shows all possible outcomes of an event.

7.01 Questions From Lesson ______________________________________________________________________ Slide 2 ⬇ What is the Probability that Benjamin wins? Recall that probability is equal to Number of favorable outcomesNumber of total possible outcomes . In this situation, a favorable outcome is (heads, heads) or (tails, tails). This makes two favorable outcomes. There are four outcomes in the sample space. The probability that Benjamin wins is equal to: Number of ways for Ben to winTotal outcomes in the sample space=2/4=1/2=0.50=50%. There is a 50% probability that Benjamin gets to sit in the front seat. What is the probability that Donna wins? Both Benjamin and Donna have a 50% chance of winning and sitting in the front seat, so this is a fair method for making this decision. ______________________________________________________________________ Slide 3 ⬇ What is the probability that both spins result in the same color? In this situation, a favorable outcome is (red, red), (yellow, yellow), (green, green), or (blue, blue). This makes four favorable outcomes. The probability that the spinner will land on the same color for both spins is equal to: Number of favorable outcomesNumber of total possible outcomes = Number of ways to get the same colorNumber of total possible outcomes = 416 = 14 =0.25=25%. There is a 25% probability that the spinner will land on the same color twice. ______________________________________________________________________ Slide 6 ⬇ Benjamin and Allen have a choice between regular soda and diet soda, and between fries and onion rings. Complete the table to create the sample space representing the choices. Allen, Benjamin, Carly, and Donna are planning to take surfing lessons. Only one person can take a lesson at a time. Use an organized list to determine the probability that they take their lesson in alphabetical order. Carly and Donna decide to get some ice cream. They have two flavor choices, chocolate or vanilla, and two choices for toppings, sprinkles or nuts. Create a tree diagram to represent the sample space

7.02 Random Lesson Stuff _________________________________________________________________________________________ Slide 1 ⬇ Allen, Benjamin, Carly, and Donna are having so much fun, they decide to stop for some frozen yogurt. Today is twisty-taste day at the yogurt shop. They will pay half-price, but the catch is that they have to take what they get. The yogurt machine will randomly give chocolate, vanilla, or cake batter flavor. They also get one free topping selected at random from the four toppings that are available. What is the probability that someone gets cake batter flavor and rainbow sprinkles? In this lesson you will follow the four friends as they calculate probabilities of compound events. _________________________________________________________________________________________ Slide 2 ⬇ The yogurt machine will randomly give chocolate, vanilla, or cake batter flavor. They also get one free topping selected at random from the 4 choices. What is the probability that someone gets cake batter flavor and rainbow sprinkles? The yogurt flavor and toppings choices are an example of a compound event. Start by making a table, an organized list, or a tree diagram of the sample space. From the tree diagram, cake batter with sprinkles is one out of the twelve possible outcomes, so the probability of getting cake batter with sprinkles is 112. It isn't practical to create a tree diagram to find probability every time. An easier method is to multiply the probabilities of each independent event. An Easier Method The yogurt flavor is independent of the toppings flavor because the outcome of the yogurt flavor does not affect the outcome of the topping. The probability of getting cake batter flavor is 13 because there is one cake batter flavor and there are three flavors all together. The probability of getting sprinkles is 14 because there is one topping choice of sprinkles and there are four toppings all together. The probability of getting cake batter flavor and sprinkles can be found by multiplying the independent probabilities together. Probability of cake batter • probability of sprinkles = 13 • 14 = 112 . The probability of getting cake batter with sprinkles is 112 Rule To find the probability of a compound event, multiply the probability of each independent event. _________________________________________________________________________________________ Slide 3 ⬇ Carly decides to go to the surf shop, which is next door to the yogurt shop, to purchase some sunscreen. When she enters the shop, Carly discovers that it's crazy discount day! When she goes to the register to pay, she has to pick a card from a regular deck of cards. If she gets a red face card, she gets a 60% discount! What is the probability that Carly gets a 60% discount? _________________________________________________________________________________________ Slide 4 ⬇ Benjamin is ordering pizza. He has three sizes to choose from: small, medium, and large. He has seven toppings to choose from, including pepperoni. If Benjamin randomly chooses a size and topping, what is the probability that he will end up with a large pizza with pepperoni? Benjamin has three sizes to choose from: small, medium, and large. The probability that he chooses a large pizza is 13 . He has seven toppings to choose from. The probability that he chooses pepperoni is 17 . The probability that he chooses a large pizza with pepperoni is: 13 • 17 = 121 . Donna has four shirts and five skirts to choose from. Two shirts are her favorite and three of the skirts are her favorite. If Donna randomly picks out a shirt and skirt without looking, what is the probability that the items are her favorites? Two of the four shirts are her favorites. The probability that she picks a favorite shirt is 24 . Three of the five skirts are her favorites. The probability that she picks a favorite skirt is 35 . The probability that she picks a favorite shirt and a favorite skirt is: 24 • 35 = 620 = 310 =0.30 = 30%. There is a 30% chance that Donna will pick two favorites. A box contains different-colored marbles that are all the same size. The table below shows the colors and amount of each marble in the box: Allen selects a marble from the box randomly, without looking, and then tosses a fair coin. What is the probability that Allen will select a red marble and the coin will land heads up? There are four red marbles and twenty total marbles. The probability of picking a red marble is 420 . The probability of the coin landing heads up is 12 . The probability of picking a red marble and the coin landing heads up is: 420 • 12 = 440 =0.10 = 10%. _________________________________________________________________________________________ Slide 5 ⬇ Summary To find the probability of a compound event, multiply the probability of each independent event. Example: There are 4 red marbles in a box and 20 total marbles. The probability of picking a red marble is 420 . The probability of the coin landing heads up is 12 . The probability of picking a red marble and the coin landing heads up is: 420 • 12 = 440 =0.10 = 10%.

7.02 Questions From Lesson

7.03 Random Lesson Stuff _________________________________________________________________________________________ Slide 1 ⬇ While Allen, Benjamin, Carly, and Donna were enjoying their day at the beach, they met a family that had four children who were all girls. "What is the probability of that happening," Donna thought? In this lesson you will learn how to design and use different simulations to generate probabilities for compound events. _________________________________________________________________________________________ Slide 2 ⬇ What is the probability that a family has four girls? You could count all the families who have four children and then determine which of those have only girls, but that's not very practical and it would take a long time. In this case, it's better to design and use a simulation to generate experimental probabilities. A family has a 50% chance of having either a boy or a girl. Having two girls is a compound event of (girl, girl). The theoretical probability of having two girls is equal to: 12 • 12 =( 12 )2= 14 = 0.25 = 25%. The theoretical probability of having four girls is equal to: 12 • 12 • 12 • 12 =( 12 )4= 116 = 0.0625 = 6.25%. A convenient way to simulate the experimental probability of having four girls is to record the results of four coins being tossed. A tossed coin has a 50% chance of landing heads up or tails up. Let "heads up" represent having a girl. Then, the proportion of coin tosses that results in four heads showing represents the experimental probability of having four girls. In this simulation, 59 of the 1000 coin tosses resulted in four heads showing. This is 591000 = 0.059 = 5.9%. From this simulation, there is a 5.9% chance that a family with four children will have all girls. That is pretty close to the theoretical probability of 6.25%. Simulation: A mathematical model used to represent a situation. Theoretical probability: Number of favorable outcomesNumber of total possible outcomes Experimental probability: Number of times an outcome occursTotal number of times the experiment is completed _________________________________________________________________________________________ Slide 3 ⬇ Suppose two friends are playing a game where they roll two number cubes to see who gets to go first. In the video, the probability that Raj and Mary had a tie was 16 , the probability that Raj won was 512 , and the probability that Mary won was 512 . If the friends didn't have number cubes, they could use other methods to simulate the same probabilities. In this spinner, two of the twelve sectors are white. The probability that this spinner will land on a white sector is 212 or 16 . This is the same proportion as the probability of rolling two number cubes and showing the same number on both cubes. The five red sectors represent the red number cube showing a greater number than the blue cube. The five blue sectors represent the blue number cube showing a greater number than the red cube. By using this spinner, the two friends have the same probability of winning as they did when rolling two number cubes. When the friends rolled two number cubes, there were 36 total outcomes. One way to represent the 36 outcomes is to use 36 colored candies in a bag. There are 6 white candies, 15 red candies , and 15 blue candies in the bag. To simulate the probabilities so they are the same as rolling the number cubes, they have to separate the candies. 636 or 212 of the candies are white which represents a tie. 1536 or 512 are red candies, which represents the red number cube showing a greater number. 1536 or 512 are blue candies, which represents the blue number cube showing a greater number. _________________________________________________________________________________________ Slide 4 ⬇ Vocabulary Simulation: A mathematical model used to represent a situation. Summary When designing a simulation, it is important to use a model that mirrors the theoretical probability. Example: To simulate the chance of a family having two boys or two girls, flip two coins and record the number of times the two coins show the same face (two heads or two tails).

7.03 Questions From Lesson ______________________________________________________________________