Unit 11 Probability. Round 1 Round 2 Round 3 Round 4


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1 Study Notes 11.1 Intro to Probability Unit 11 Probability Many events can t be predicted with total certainty. The best thing we can do is say how likely they are to happen, using the idea of probability. Activity: Play the game Greed. Rolling one dice if a 3 is rolled you re out of the game. Round 1 Round 2 Round 3 Round 4 Total: Total: Total: Total: Overall Total Example 1: List the sample space for each. Sample Space: The set of all possible outcomes of an experiment. A. Rolling a die once B. Flipping a coin C. Flipping a coin twice In general, probability is the ratio of the number of ways the outcome can happen to the number of possible outcomes. Example 2: Using the sample spaces from Example 1, determine the probability that each event occurs. A. Rolling a 3 on a six sided die. B. Getting a head and then a tail when a coin is flipped twice. Activity: Play the game Greed. Rolling two dice if the sum of the dice is 7 you re out of the game. Round 1 Round 2 Round 3 Round 4 Total: Total: Total: Total: Overall Total
2 Example 3: List the sample space for the sum of two dice. A. What is the probability of a sum of 7? B. Sketch a histogram of the frequencies. C. After calculating the probability, and viewing the histogram of the frequencies. Would your strategy for playing Greed change? If so, how so? Example 4: If one of the blocks is chosen at random, what is the probability that it will have a vowel on it? Example 5: For this spinner, what is the probability of the spinner landing on a shaded section? Event: a single result of an experiment. Complement of an Event: all outcomes that are NOT the event. Notation: If E is the event, then the complement is written or. 1 Example 6: For the blocks above, what is the complement of picking a vowel? For the spinner above, what is the complement of the spinner landing on a shaded section? What is the probability of the complement?
3 Example 7: Use the box below to answer the following questions. Assume a ball is chosen at random. A. B. C. D. E. Dependent Events: the outcome of the experiment changes based on an outcome that previously happened. For example: You have 2 red, 1 blue, and 3 green socks in a drawer. Suppose you reach into the drawer without looking, choose a sock, and then reach in again and choose another sock. What is the probability of choosing a red sock the first time? What is the probability of choosing a red sock the second time? These probabilities are different, so we know the events are dependent. Independent Events: the outcome is not affected by previous events. For example: You have 5 red, 2 orange, and 7 black marbles in a bag. Suppose you reach into the bag without looking and pick up a marble, look at the color, and then put it back in the bag. You reach in again to pull out another marble What is the probability of choosing a black marble the first time? What is the probability of choosing a black marble the second time? These probabilities are the same, so we know the events are independent. Example 8: Determine whether the events are independent or dependent. Explain your reasoning. A. Tossing a coin multiple times. For example, you toss a coin and it comes up heads three times. What is the probability that the next toss will also be a heads? B. Picking cards from a deck, one at a time. Is the event picking a face card independent or dependent? Compound Event: An event that consists of two or more events. We can calculate the chance of two or more independent events by multiplying the probability of each event. Example 9: You toss a coin 3 times. What is the probability that a head comes up all three times? Example 10: What is the probability of randomly choosing a shaded block from group 1 and then choosing a cylinder from group 2?
4 But what if the events are dependent? The probability that Event A happens and Event B happens is the product of the probability that Event A happens and the probability that Event B happens given that Event A has already happened. Example 11: Suppose you have 5 red, 2 orange, and 7 black marbles in a bag. You reach in without looking and choose a marble, do NOT replace it, and then choose another marble. A. Why are these events dependent? B. What is the probability of choosing a red marble and then an orange marble? C. What is the probability of choosing a red marble and then a marble that is not black? D. What is the probability of choosing two red marbles? OR probabilities Example 12: Use the box below to answer the following questions. Assume that a shape is chosen randomly. A. B. C. D. Example 13: A number cube has a different number on each of 6 sides. What is the probability of rolling a number cube twice such that you roll a 5 the first time or a 6 the second time? Event A: Event B: Example 14: Terrance has different colors of tee shirts in his drawer: 2 red t shirts, 2 green t shirts, and 3 blue t shirts. If he chooses a t shirt at random from the drawer without replacing it, and then randomly chooses another, what is the probability of choosing a blue shirt first OR a green shirt second?
5 Extra Practice: A standard deck of cards contains 13 (numbered 2 10, Jack, Queen, King, Ace) cards in each of the four suits (Hearts, Clubs, Spades, Diamonds) making 52 cards total. Example 15: Alex draws 2 cards from the deck, but does not replace the first before drawing the second. Find the probability of each of the following events. As you answer each question make sure to use correct notation. A. Is Alex drawing 2 cards from the deck and not replacing the first before drawing the second an example of dependent or independent events? Explain why? B. What is the probability Alex draws two red cards? C. What is the probability Alex draws a red card then a black one? D. What is the probability that Alex draws a two then a face card? Example 16: Dallas draws 2 cards from the deck, but replaces each one before drawing the next. Find the probability of each of the following events. As you answer each question make sure to use correct notation. A. Is Dallas drawing 2 cards from the deck and then replacing each one before drawing the next an example of dependent or independent events? Explain why? B. What is the probability of Dallas drawing two red cards? C. What is the probability of Dallas drawing a heart and a 5? D. What is the probability of Dallas drawing a spade or a face card? E. What is the probability of Dallas drawing a black card or a 6? F. What is the probability of Dallas drawing a 3 for the first card but then not drawing the queen of spades for the second?
6 Study Notes 11.2 TwoWay Tables, Long Run Probability (Law of Large Numbers) Review: Find the indicated probability. Independent: Dependent: A. 0.4, 0.35, 0.5,?? B. 0.6, 0.2,??, 0.1 C. 0.25,??, 0.70, 0 There is many different ways to visualize data. Each method has its strengths and weaknesses. We will be focusing on Two Way Tables, Venn Diagrams and Tree Diagrams. TwoWay Tables A two way frequency table shows the number of data points and their frequencies for two variables. One variable is divided into rows and the other is divided into columns. Two way frequency tables can be used to calculate probabilities of compound events. Example 1: This two way frequency tables show how many 9 th and 10 th graders are studying Spanish, French, or German. A. What is the probability that a randomly selected student is studying French? B. What is the probability that a randomly selected student is a 9 th grader? C. What is the probability that a randomly selected 10 th grader is studying German? D. What is the probability that a randomly selected Spanish student is a 9 th grader? E. What is the probability that a randomly selected 9 th grader is studying French? F. Are the events of studying Spanish and being a 9 th grader independent or dependent?
7 Conditional Probability: The probability that an outcome occurs, given that another event has already occurred. Example 2: Complete the two way table to show the distribution of member of the audience at a play. Then answer the questions that follow. A. How many total people were in the audience? B. What is the probability that a randomly selected audience member sat in the balcony? C. What is the probability that a randomly selected audience member was an adult? D. What is the probability that a randomly selected audience member was a child who sat in the stalls? E. What is the probability that a randomly selected audience member sat in the balcony or was an adult? F. What is the probability that a randomly selected child sat in the circle? G. What is the probability that a randomly selected circle sitter was a child? Example 3: Use the information provided to make a two way table. Data was collected at the movie theater last fall. Not about movies but clothes. 6,525 people were observed 3,123 had on shorts and the rest had on pants 45% of those wearing shorts were denim Of those wearing pants 88% were denim. Total After completing the table, write three conditional statements (using correct notation) and provide the answer. A. B. C. Total 6,525 Activity: Spinning in Circles
8 Study Notes 11.3 Venn & Tree Diagrams Review: Last class period we ended with the activity Spinning in Circles. With a partner summarize in 3 4 sentences what we learned from the activity. We discovered that Probabilities can be interpreted as the long run relative frequencies of outcomes. That is, a probability can be interpreted as the proportion of time that an outcome will occur in the long run, over many trials. This is also known as the Law of Large Numbers. The law of large numbers is the principle of probability to which the frequencies of events with the same likelihood of occurrence even out, given enough trials or instances. As the number of experiments increases, the actual ratio of outcomes will converge on the theoretical, or expected, ratio of outcomes. Below is an example of data that is collected over time, so the estimated probability outcome becomes more precise as the sample size increases over time. Example 1: Freddy loves fried food. His passion for the perfect fried food recipes led to him opening the restaurant, Fried Freddie s. His two main dishes are focused around fish or chicken. Knowing he also had to open up his menu to people who prefer to have their food grilled instead of fried, he created the following menu board: After being open for six months, Freddy realized he was having more food waste than he should because he was not predicting how much of each he should prepare in advance. His business friend, Tyrell, said he could help. A. What information do you think Tyrell would need? Luckily, Freddy uses a computer to take orders each day so Tyrell had lots of data to pull from. After determine the average number of customers Freddy serves each day, Tyrell had the following data. On average, Freddy serves 300 customers a day. 35% of his customers preferred fried food 45% of his customers preferred fish 15% of his customers preferred fried fish B. Create a two way table displaying the data. Total Total 300
9 C. Let s now try representing the data in a different way, use the Venn diagram below to represent what Freddy knows about his customers. Venn diagram: A Venn diagram represents values as circles with an enclosing rectangle, common elements of the sets being represented by the areas of overlap among the circles. Fried Fish To make sense of the diagram, Freddy computed the following probability statements: D. What is the probability that a randomly selected customer would order fish? Shade the part of the diagram that models this situation. E. What is the probability that a randomly selected customer would order fried fish? Shade the part of the diagram that models this solution. F. What is the probability that a person prefers fried chicken? Shade the part of the diagram that models this solution. G. What is the probability that a randomly selected person would choose fish or fried? Shade the part of the diagram that models this solution. H. What is the probability that a randomly selected person would NOT choose fish or fried? Shade the part of the diagram that models this solution. I. If Freddy serves 120 meals at lunch on a particular day, how many orders of fish should he prepare with his famous fried recipe? Think About It: Sally was assigned to create a Venn diagram to represent. A. Sally first writes, what does this mean? Explain each part in words. B. Sally then creates the following diagram. Sally s Venn diagram is incorrect. Why?
10 The Two way table from Example 1b. Fried Grilled Total Chicken Fish Total Using the data in the table let s represent Freddy s data a third way. With a tree diagram. Tree diagram: A Tree diagram is used in strategic decision making, valuation or probability calculations. The diagram starts as a single node, with branches leading to different possible occurrences. Which represent mutually exclusive decisions or events. Fried Reminder: Probabilities need to add to 1 Grilled A. Based on the data, if someone likes fried food what is the probability they prefer chicken over fish? B. Based on the data, if someone likes grilled food what is the probability they prefer fish over chicken? C. What is the probability that someone likes fried fish? D. What is the probability that someone likes grilled fish? E. What is the probability that someone likes chicken? F. What is the probability that someone likes fish?
11 Example 2: Danielle loves chocolate ice cream much more than vanilla and was explaining to her best friend Raquel that so does most of the world. Raquel disagreed and thought vanilla is much butter. To settle the argument, they created an online survey asking people to choose their favorite ice cream flavor between chocolate and vanilla. After completing the survey, the following results came back: There were 8,756 females and 6,010 males who responded. Out of all the males, 59.7% chose vanilla over chocolate. 4,732 females chose chocolate. A. Upon first observations, which flavor do you think won? Write a sentence describing what you see at first glance that makes you think this. B. Raquel started to organize the data in the following two way table. See if you can help complete this (using counts and not percentages): Chocolate Vanilla Total Female 8,756 Male 6,010 Total Now organize the data using a Venn diagram and a Tree diagram. A. Using your organized data representations, write probabilities that help support your claim regarding your preferred flavor of ice cream. For each probability, write a complete statement as well as the corresponding probability notation. (You should calculate a minimum of 3 probabilities) B. Looking over the three representations (tree diagram, two way table, and Venn diagram), discuss the strengths and weaknesses of each. Tree Diagram Strength Tree Diagram Weakness Two way Table Two way Table Venn Diagram Venn Diagram
12 Study Notes 11.4 Independence Review: Determine whether the events are independent or dependence of each other. Explain. A. Rolling a six sided die, then drawing a card from a deck of 52 cards. B. Drawing a card from a deck of 52 cards, then drawing another card from the same deck without replacement. C. Rolling a six sided die, then rolling it again. D. Pulling a marble out of a bag, replacing it, then pulling a marble out of the same bag. E. Having 20 treats in five different flavors for a soccer team, with each player taking a treat. The definition of independence is that two events (A and B) are said to be independent if one of the following occur: Determining the independence of events can sometimes be done by becoming familiar with the context in which the events occur and the nature of the events. Or we can determine independence of events based on equivalent probabilities. Example 1: Using the definition of independence determine if events A and B are independent. A.,, B.,, In the next few examples use your knowledge of conditional probability as well as the definition of independence to answer the following questions. Keep track of how you are determining independence for each type of representation.
13 In each of the Venn Diagrams the number of outcomes for each event are given, use the provided information to determine the conditional probabilities or independence. The numbers in the Venn diagram indicate the number of outcomes in that part of the sample space. Example 2: A. How many total outcomes are possible? B. C. D. E. F. G. Are events A and B independent events? Why or why not? Example 3: A. How many total outcomes are possible? B. C. D. E. F. G. Are events E and F independent events? Why or why not? Example 4: A. How many total outcomes are possible? B. C. D. E. F. G. Are events X and Y independent events? Why or why not?
14 Example 5: Out of 2000 students who attend a certain high school, 1400 students own cell phones, 1000 own a tablet, and 800 have both. Suppose a student is randomly selected. Create a Venn diagram and use proper notation to answer the following questions. A. What is the probability that a randomly selected student owns a cell phone? B. What is the probability that a randomly selected students owns both a cell phone and a tablet? C. If a randomly selected student owns a cell phone (was one of the 1400 with a phone), what is the probability that this student also owns a tablet? D. How are questions (B) and (C) different? E. Are the outcomes owns a cell phone and owns a tablet independent? Explain. F. If question e is not independent, what number of students would need to own a tablet to create independence?
15 Example 6: Data gathered on the shopping patterns during the month of April and May of high school students from Peanut Village revealed the following. 38% of students purchased a new pair of shorts (call this event H), 15% of students purchased a new pair of sunglasses (call this event G) and 6% of students purchased both a pair of shorts and a pair of sunglasses. A. Using a table organize the information provided. B. Find the probability that a student purchased a pair of sunglasses given that you know they purchased a pair of shorts. C. Find the probability that a student purchased a pair of shorts or purchased a new pair of sunglasses. D. Given the condition that you know a student has purchased at least one of the items. What is the probability that they purchased only one of the items? E. Are the two events H and G independent of one another? Why or why not? Think About It: Explain what independence looks like in a Venn diagram, a tree diagram, and a two way table. (Your answer must include pictures and text) Two Way Table Venn Diagram Tree Diagram
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