Math 7, Unit 5: Probability - NOTES

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1 Math 7, Unit 5: Probability - NOTES NVACS 7. SP.C.5 - Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability round ½ indicates an event that is neither likely nor likely, and a probability near 1 indicates a likely event. Most of us would like to predict the future; just think of the possibilities if we could! Since we cannot, the best we can do is tell how likely something is to happen. It s helpful to know if something is impossible, likely, unlikely, or certain to happen. People like to know if it is a sure thing, or a chance or it will never happen. It is more useful if you can use a number to describe the likelihood. Probability is the measure of how likely an event is to occur. They are written as fractions or decimals from 0 to 1, or as a percent, 0% to 100%. The higher the probability, the more likely an event is to happen. For instance, an event with a probability of 0 will never happen. If you have a probability of 100%, the event will always happen. An event with a probability of 1 or 50% has the same chance of 2 happening as not happening. Let s take this example. Two of five unmarked envelopes contain $100 bills. The other three are empty. How likely is it that you will choose an envelope with a bill in it? To calculate the probability, you need to know all the different things that can happen. A sample space is a list of all the possible outcomes of an event. When you pick one of the envelopes, the result is called an outcome. Since there are 5 envelopes, there are 5 possible outcomes or the sample space is envelope 1, envelope 2, envelope 3, envelope 4, envelope 5. Each of these are equally likely and 2 of them are favorable. We say that the probability of drawing an envelope with a bill in it is 2, 0.4 or 40%. Using a shorthand, if we let E stand 5 for the event of drawing an envelope with a bill in it, we can write P(E)= 2 to mean that the probability of 5 event E is 2. We could say the event is unlikely to happen. 5 Example: How likely is it to pick a black ball from a bag with four red balls in it? In this case, you definitely could not pick a black ball; so it is impossible you would draw a black ball. The probability would be 0 because it would never happen. Example: How likely is it to pick a red ball from a bag with four red balls in it? In this case, you would definitely pick a red ball; so it is certain you would draw a red ball. The probability would be 1 because it would happen each and every time or 100% of the time. Example: How likely is it to pick a red ball from a bag with four red balls in it? In this case, you would definitely pick a red ball; so it is certain you would draw a red ball. The probability would be 1 because it would happen each and every time or 100% of the time. Example: How likely is it to pick a red ball from a bag with one red ball and one black ball? In this case, you might pick a red ball or you might pick a black ball. The chance is or equally likely and the probability would be ½. Math 7 Notes Unit 5: Probability Page 1 of 36

2 Example: How likely is it to pick a red ball from a bag with two red balls and two black balls in it? Again you might pick a red or a black ball and the chances are equally likely or 2 4 which is ½. Example: How likely is it to pick a red ball from a bag with two black balls and eight red balls? Even though it is not certain that you will get a red ball, a red ball would be selected most of the time because there are many more reds than blacks. So we say this is likely. This time the probability is less than 1 but greater than ½. In fact since we have eight chances out of ten to pick a red ball, we say the probability is 8 = 10 4 or 80%. 5 Example: How likely is it to pick a red ball from a bag with eight black balls and two red balls? Even though it is not certain that you will get a red ball, a red ball would NOT be selected most of the time because there are many fewer reds than blacks. So we say this is unlikely. This time the probability is less than ½. but greater than 0 In fact, since we have two chances out of ten to pick a red ball, we say the probability is 2 = 1 or 20% The figure below shows a probability scale: x Impossible Unlikely Equally Likely to Occur or Not Occur Likely Certain Example: How likely is it that a coin tossed will come up heads? There is as likely a chance of heads as not heads. In other words, a probability of 1 or 0.5 or 50%. The chances are equally likely. 2 Example: How likely is it that a tossed die (number cube) will be a prime number? The prime numbers on a die would include 2, 3 and 5. Die could have a total of 6 possibilities (1, 2, 3, 4, 5, 6), so the probability of a prime number would be: number of favorable outcomes probability = = 3 = 1 = 50% so it has the same chance of number of possible outcomes 6 2 happening or not happening. Example: How likely is it that a tossed die (number cube) will be a less than 7? The numbers on a die less than 7 would include 1, 2, 3, 4, 5, and 6. Die could have a total of 6 possibilities (1, 2, 3, 4, 5, 6), so the probability of a number less than 7 would be: number of favorable outcomes probability = = 6 = 1 = 100% so it is a sure thing. number of possible outcomes 6 1 Example: How likely is it that a tossed die (number cube) will be a less than 2? The numbers on a die less than 2 would include only 1. Die have a total of 6 possibilities (1, 2, 3, 4, 5, 6), so the probability of a number less than 2 would be: Math 7 Notes Unit 5: Probability Page 2 of 36

3 number of favorable outcomes probability = = 1 = 0.16 = 16 2 % so it is unlikely. number of possible outcomes 6 3 Example: How likely is it that a single tossed die (number cube) will be an 8? The numbers on a die would include 1, 2, 3, 4, 5, and 6. There is no 8 on a single die, so the probability of an 8 would be: number of favorable outcomes probability = = 0 = 0 = 0% so it will never happen. number of possible outcomes 6 Example: The weather report gives a 75% chance of rain for tomorrow. This means that there is a likely chance of rain (75%) and an unlikely chance of no rain (25%). In other words, the probability of rain is 3 or 0.75 or 75%. 4 Example: How likely is it that a single tossed die (number cube) will not be 4? The numbers on a die would include 1, 2, 3, 4, 5, and 6. To not be a 4, the number could be 1, 2, 3, 5 or 6. number of favorable outcomes probability = = 5 =.83 = 83 1 % so it should happen; it s number of possible outcomes 6 3 very likely or more likely to happen. Example: How likely is it that a single coin tossed will be heads? Tossing a coin; the possible outcomes (results) are a head or a tail. number of favorable outcomes probability = = 1 = 0.5 = 50% so it would be equally number of possible outcomes 2 likely to occur as not to occur. Theoretical probability is based on knowing all the equally likely outcomes of an experiment, and it is defined as a ratio of the number of favorable outcomes to the number of possible outcomes. Outcomes are the possible results of an experiment. Mathematically, we write: or number of favorable outcomes probability = number of possible outcomes success probability = success + failure Example: Suppose you pick a marble from a hat that contains three red, two yellow and one blue marble. What is the probability you draw a yellow marble? # of yellow marbles P( yellow marbles ) = total number of marbles Math 7 Notes Unit 5: Probability Page 3 of 36

4 2 1 P = or 6 3 The probability found in the above example is an example of theoretical probability. Example: Shade each spinner below (with the words impossible, unlikely, equally likely to occur or not to occur, likely and certain) to describe the chance of the spinner landing on black. equally impossible unlikely likely likely certain Example: In words, describe a situation where the outcome would be: a. unlikely b. impossible c. certain d. equally likely e. likely Experimental probability is the ratio of the number of times an event occurs to the total number of trials based on repeated trials of an experiment. Example: In the last thirty days, there were 7 cloudy days. What is the experimental probability that tomorrow will be cloudy? 7 P( cloudy days ) = = Example: Marco attempted 28 free throw and made 17 of them. Her experimental probability of making a free throw is 17 =.60 5 = 60 5 % Example: The four sections of a spinner are colored red, yellow, green and blue. The arrow on the spinner is spun 10 times. The color in the section where the arrow stops for each spin is listed below. red blue red yellow red red green red blue red yellow green blue red The arrow on the spinner is spun again. What is the experimental probability that the arrow stops on a section colored red? A. 25% B. 40% C. 60% D. 75% Math 7 Notes Unit 5: Probability Page 4 of 36

5 Example: The arrow on the spinner at right will be spun 400 times. Based on theoretical probability, which graph best represents the number of spins, in 400 spins, that the arrow will stop in each color section? red red blue blue white A. C. Theoretical Results Theoretical Results Number of Spins Red White 1 Blue Section Color Column1 Column2 Column3 Column4 Number of Spins Red White Blue Section Color Series 1 Series 2 Series 3 series 4 series5 B. D. Theoretical Results Theoretical Results Number of Spins Red White Blue Section Color Number of Spins Red White Blue Section Color Series 1 Series 2 Series 3 Series 4 Series 5 Example: Find the probability of picking a striped card. (Use graphic below.) Then tell whether the event is impossible, unlikely, likely, or certain. Math 7 Notes Unit 5: Probability Page 5 of 36

6 Example: In a school raffle, there is a 40% chance of winning a small prize, a 20% chance of winning a large prize and a 40% chance of winning no prize. Is it more likely to win a small prize or to win no prize? A. The person is more likely to win no prize. B. The person is more likely to win a small prize. C. The person is as likely to win a small prize as to win no prize. Example: There is a 50% chance that someone reaching in a bag of marbles will draw out a purple marble, a 20% chance they will draw out a green marble, and a 30% chance they will draw out a red marble. Is it less likely that a person will draw a green marble than a purple marble? Explain your thinking. Example: At a carnival game, you may win an inflatable crayon, you may win a small stuffed animal, or you may win nothing at all. If the probability of winning nothing is 0.63 and the probability of winning a small stuffed animal is 0.27, what is the probability of winning an inflatable crayon? Express your answer as a decimal. Example: A bag contains blue, yellow and re marbles. The probability of drawing a yellow marble is 3. What is the probability of NOT drawing a yellow 20 marble? Looking for more examples and/or readymade worksheets on this standards? Click on the link below. NVACS 7. SP.C.6 - Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. A chance event like the ones described below, flipping a coin 5 times, or selecting a cube from a bag of 10 cubes are examples of a chance event. An estimate for the probability of a chance The number of times you win event is = The total number of times you play the game or Favorable outcomes. Total number of outcomes Students need to experiment or play with different scenarios. One example would be to divide students into groups of 2. Give each group a number cube and a chart like the one below. Each group will play the game 15 times and record each outcome of each spin in the table below. Once the table is complete, they will answer the questions below. Turn 1 st rolls result 2 nd rolls result Sum of rolls 1 and Math 7 Notes Unit 5: Probability Page 6 of 36

7 Out of the 15 turns how many times was the sum greater than 6? 2. What sum occurred the most often? 3. What sum occurred the least often? 4. If a lot of games, let s say 150 games were played, what portion of the games will they win? Explain your answer. 5. Name a sum that would be impossible to get when playing the game. 6. What event is certain to occur while playing the game? 7. Based on your experiment of the game, what is the estimate for the probability of getting a sum of 7 or more? 8. Based on your experiment of playing the game, what is the estimate for the probability of getting a sum of exactly 8? ***A variation of this could be done with spinners. Given a spinner with 4 equal sections numbered 1 4. Students could spin the spinner twice for each of the 15 turns. Small bags of M&M s (using colors) or small boxes of Animal Crackers (using specific animals) would provide additional variations on developing this skill. Teachers may want to total all the teams outcomes to compare the individual group results with the entire classes. This would help students develop the understanding that the more outcomes they determine, the more confident they can be that the proportion of winning the game is providing an accurate estimate of the probability. You may also want to sum all class results at the end of the day to compare even more outcomes to each individual group result the next day. Example: Yeji tossed a paper cup 40 times and recorded how the cup landed each time. He organized the results in a table shown below. Find the experimental probability that the cup will NOT land right-side up. Express your answer as a fraction in simplest form. Outcome Right-side up Upside down On it side Frequency Math 7 Notes Unit 5: Probability Page 7 of 36

8 Example: Jesse tossed a paper cup 30 times and recorded how the cup landed each time. He organized the results in a table shown below. Based on the results, is the cup more likely to land right-side up or on its side? Outcome Right-side up Upside down On it side Frequency Example: If you roll a number cube 72 times, how many times do you expect to roll a 6? Example: An airplane flight has 228 seats. The probability that a person who buys a ticket actually goes on the flight is about 95%. If the airline wants to sell all the seats on the flight, how many tickets should they sell? Example: Kia s experimental probability of striking out at baseball is 13%. Out of 30 times at bat, about how many times will she strike out? Looking for more examples and/or readymade worksheets on this standards? Click on the link below. NVACS 7. SP.C.7 - Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observe frequencies; if the agreement is not good, explain possible sources of the discrepancy. mexample: Example: Jennifer brings a bag to school filled with colored bracelets that she made. The table gives the number of each colored bracelet in the bag. Color Frequency green 2 blue 10 red 8 purple 5 Math 7 Notes Unit 5: Probability Page 8 of 36

9 Part A: Create a probability model for each color of bracelet. Part B: Use the probabilities to predict the number of red bracelets that would be drawn in a random selection of 10 bracelets. Part C: Simulate a drawing of 10 randomly selected bracelets. How do the results compare with your prediction? Explain possible sources of any discrepancy. Example: An independent polling company conducted a survey with a random sample of 500 people to find out which type of vehicle they preferred to own. The results are shown below. Vehicle Number of People Compact 75 Minivan 45 Pickup 95 Sedan 100 SUV 95 Sport 90 Part A: What is the probability that a randomly selected survey participant prefers to own a sedan. Explain. Part B: If 1,000 people were polled how many would you expect to prefer to own an SUV? Explain. Part C: Marcus conducted the same survey of 200 high school seniors. His results are shown below. Vehicle Number of People Compact 35 Minivan 15 Pickup 35 Sedan 40 SUV 50 Sport 25 If his results had been more like the survey company, how many fewer students would have picked the SUV as their favorite vehicle to own? Explain. NVACS 7. SP.C.7a -Develop a uniform probability model assigning equal probability to all outcomes, and use the model to determine probabilities of events. A uniform probability requires that all outcomes must be the same or equal. For example, rolling a number cube there are 6 possible outcomes and each one is as likely to occur. Flipping a coin is uniform because it is equally likely that you will get a heads as a tail. With a standard deck of playing cards each card in the deck is likely to occur when drawing one card. Example: What is the probability of flipping a coin and getting tails? Math 7 Notes Unit 5: Probability Page 9 of 36

10 number of favorable outcomes probability = number of possible outcomes = 1 2 Example: What is the probability of rolling a 5 with a fair number cube? number of favorable outcomes probability = number of possible outcomes = 1 6 Example: What is the probability of rolling a number less than 4 with a fair number cube? number of favorable outcomes probability = = 3 = 1 number of possible outcomes 6 2 Students need to experiment with a deck of cards and know, or come to know, that there are 52 cards in the deck, there are 26 red cards and 26 black cards, there are 4 suits (hearts, diamonds, spades and clubs), and 13 cards within and each suit. Each suit contains 4 aces, 4 twos, 4 threes, 4 fours, 4 fives, 4 sixes, 4 sevens, 4 eights, 4 nines, 4 tens, 4 jacks, 4 queens and 4 kings. Example: What is the probability of picking a 5 from a standard deck of 52 cars? number of favorable outcomes probability = = 4 = 1 number of possible outcomes Example: What is the probability of picking a red card from a standard deck of 52 cars? number of favorable outcomes probability = = 26 = 1 number of possible outcomes 52 2 When rolling two number cubes students need to experience and know, or come to know, the sample space is: There are 36 possible outcomes. Example: An experiment consists of rolling two fair number cubes. Find the probability of rolling two 3 s? Express your answer as a fraction in simplest form. number of favorable outcomes probability = number of possible outcomes = 1 36 Math 7 Notes Unit 5: Probability Page 10 of 36

11 Example: An experiment consists of rolling two fair number cubes and summing the result. Find the probability of rolling a sum of 6? Express your answer as a fraction in simplest form. number of favorable outcomes probability = number of possible outcomes = 5 36 Example: An experiment consists of rolling two fair number cubes and summing the result. Find the probability of rolling a sum of more than 6? Express your answer as a fraction in simplest form. number of favorable outcomes probability = = 21 = 7 number of possible outcomes Example: An experiment consists of rolling two fair number cubes and summing the result. Find the probability of rolling a sum of less than 10? Express your answer as a fraction in simplest form. number of favorable outcomes probability = = 30 = 5 number of possible outcomes 36 6 Example: Suppose the probability of a certain outcome in a fair experiment is 1. What is the 3 probability of the outcome NOT occurring? What can you say about the number of possible outcomes of the experiment? Explain how you know. If the probability of a a certain outcome is 1, then the probability of the outcome not occuring is 1 = because the probabilities must add to If the experiment is fair, the probability of all outcomes must be the same, so every outcome has to have a probability of 1. The probabilities must sum to 1, so there 3 must be exactly 3 outcomes, because = NVACS 7. SP.C.7b -Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. You can toss a paper cup, use a random generator for a set of numbers, pick from a deck of cards (with replacement), use a spinner, pick from a bag of tiles or marbles (with replacement), etc. to demonstrate experimental probability. Students need to experiment with this standard. Regardless of the medium you choose, do the following: 1. Discuss the possible outcomes? e.g., the cup could land on its side, open-end up or open-end down. 2. Ask students what outcome they think will most likely occur? 3. Have students write a statement about each outcome using the words likely and unlikely. Math 7 Notes Unit 5: Probability Page 11 of 36

12 4. Create a chart listing each outcome. Have students perform the simulation and record the number of times each outcome occurs. Then have students compute the experimental probability. 5. Ask students if each group will have the exact same results. 6. Have students sum the probability of all the outcomes and discuss why they sum to 1 or ask how to find the experimental probability of one of the outcomes in a different way. e.g., sum the experimental probability of the cup landing on its side and landing open ended. Then subtract that amount from 1. Another way to say that is 1 P(complement). If the outcomes are do not agree with what you thought would be the results, discuss why the results might be faulty or erroneous. e.g., the cup was not randomly throw it was purposely dropped to land in a particular way, when replacing the chosen card they were put at the bottom of the deck instead of being shuffling back into the deck, the tile was replaced to the left side of the bag and the picker drew only from the left side of the bag, etc. Example: The letters R, W, A, and E are arranged in a random order. What is the probability that the arrangement forms the words WEAR or WARE? Example: It is Erica s turn to spin in a game she is playing with her friends. A. 1 2 B. 1 6 C D What is the probability that Erika will get to move ahead on this spin? A. 5 8 B. 1 3 C. 3 8 D. 1 2 Looking for more examples and/or readymade worksheets on this standards? Click on the link below. NVACS 7. SP.C.8 - Find probabilities of compound events using organized lists, tables, tree diagrams and simulation. Tree Diagrams and The Fundamental Counting Principle One method students may use to determine the number of possible outcomes is to write an organized list. Example: Cassie has a blue sweater, a red sweater, and a purple sweater. She has a white shirt and a tan shirt. How many different ways can she wear a sweater and a shirt together? Math 7 Notes Unit 5: Probability Page 12 of 36

13 blue sweater, white shirt or BW blue sweater, tan shirt or BT red sweater, white shirt or RW red sweater, tan shirt or RT purple sweater, white shirt or PW purple sweater, tan shirt or PT 6 ways As more items are added, this method becomes cumbersome. Another method students may use to determine the number of possible outcomes, is to create a table. Example: Rolling two number cube, create and complete a table for the sum of the two numbers How many possible sums are there? What is the probability that the sum is 12? 3. What is the probability that the sum is 6? What is the probability that a sum of six is achieved rolling double threes? A tree diagram is another method for listing outcomes. This method makes it easier to see (count) the number of possible outcomes for experiments when the numbers are reasonable and there are multiple events. To draw a tree diagram, you: 1) begin with a point; then you draw a line for each outcome in the first event. 2) draw lines for subsequent outcomes based on the outcomes from the first event. Example: Draw a tree diagram to show the outcomes for flipping two coins. Start with a point. There are two outcomes for the first coin, a head (H) or a tail (T). Draw lines and label H and T. H T Math 7 Notes Unit 5: Probability Page 13 of 36

14 For either of the outcomes in the first flip, the second coin could be a head or a tail. So the tree diagram would look like this. H T H T H T Now reading down the tree diagram, the possible outcomes are HH, HT, TH, or TT. There are 4 possible outcomes when flipping two coins. Extend the tree diagram for three coins. How many outcomes are there then? Example: Draw a tree diagram to determine the number of different outfits that could be worn if you had two pairs of pants and three shirts. Starting with a point, you have 2 pairs of pants. For each pair of pants (P1 and P2), you have three shirts (S1, S2 and S3) to choose from. P1 P2 S1 S2 S3 S1 S2 S3 There are 6 possible outcomes: P1S1, P1S2, P1S3, P2S1, P2S2, and P2S3. Example: Popcorn at the movie theatre comes in three sizes small, medium and largeand it comes either plain or buttered. You can add salt or not add salt. How many choices of popcorn do you have? Show your work. with salt small without salt Plain medium large small with salt without salt with salt without salt with salt without salt Buttered medium with salt without salt large with salt without salt Math 7 Notes Unit 5: Probability Page 14 of 36

15 Fundamental Counting Principle Organized lists, tables and tree diagrams are useful to get a picture of what is occurring, but with a large number of events, each method can get out of hand in a hurry. A quick way to determine the number of possible outcomes in an event is to multiply the number of outcomes in each event. With the first example involving three sweaters and two shirts, there are 3 possible choices for a sweater, paired with each of the two shirts or 3 2 = 6. The second example involving the sum of two rolling cubes, there are six outcomes on the first die, and then when each of those is paired with 6 different possible outcomes for each of the numbers on the first die, we see 6 6 = 36. The next example, flipping coins using 2 coins, there are 2 outcomes when you flip the first coin and two outcomes when you flip the second coin. 2 2= 4. In the next example, we choose from 2 pairs of pants, then from three shirts. Notice the total number of outcomes we identified using the tree diagram was = 6. The last example involving popcorn, would yield 2 types, 3 sizes, with or without salt would yield 232 = 12choices. Those examples lead us to the following generalization: Fundamental Counting Principle: If one event can occur in m ways, and for each of these ways a second event can happen in n ways, then the number of ways that the two events can occur is m n. Example: How many possible outcomes are there if you roll two cubes with the numbers one through six written on each face? There are 6 outcomes on the first cube, 6 outcomes on the second cube, so using the Fundamental Counting Principal we have 6 6 = 36 outcomes. Example: How many possible outcomes are possible for tossing a coin and rolling a cube with the numbers one through six written on each face? There are two things that can happen when tossing a coin. There are six things that can happen when rolling the cube. Using the Fundamental Counting Principle, we have 2 6 = 12 outcomes. Example: How many possible answers are there to a 10 question True-False test? Using Fundamental Counting Principal, = 2 = Math 7 Notes Unit 5: Probability Page 15 of 36

16 Example: The combination for the locks on gym lockers is a three-digit code using the digits 0, 1 and 2. How many different combinations are possible? Make an organized list, table, or tree diagram and then use the Fundamental Counting Principle to verify your results. Example: A deli has 5 different types of bread (white, wheat, rye, pumpernickel and sourdough), and 7 types of meat (ham, roast beef, turkey, bologna, tuna, salami and prosciutto). If you choose 1 bread and 1 meat, how many possible different sandwiches can be made? Make an organized list, table, or tree diagram and then use the Fundamental Counting Principle to verify your results. NVACS 7. SP.C.8a - Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs. Let s extend this to finding the probability of compound events (an event made up of two or more separate events). If the occurrence of one event does not affect the probability of the other, the events are independent. An example of independent events would be rolling a die and spinning a spinner. Another might be flipping a coin and drawing a marble from a bag. Yet another might be rolling a 1 and a 2 when rolling two different number cubes, or flipping one coin multiple times. If the occurrence of one event does affect the probability of the other, the events are dependent. An example of dependent events would be drawing from a bag of marbles then drawing a second marble without replacing the first marble. Another example could be choosing a book off a shelf then the next person choosing a book of the same shelf without replacing the first book. One might choose a CD off a pile and then choose a second one from the pile without replacement (not returning the first CD before the second is chosen). Probability of Independent Events = P(A) P(B) Example: An experiment consists of flipping a coin 2 times. What is the probability of flipping heads both times? The flip of a coin does not affect the results of the other flips, so the flips are independent. For each flip, P(H) = So P(H, H) = or Example: You have 3 colors of t-shirts (red, blue, green) 2 colors of shorts (white, black) from which to choose. What is the probability of randomly choosing a blue shirt with black pants? For the choice of t-shirt, P(T) = 1 3, for the shorts P(S) = ; So P(T, S) = or Math 7 Notes Unit 5: Probability Page 16 of 36

17 Example: To unlock a car door requires a 4-digit code on the keypad. The code uses the numbers 0 and 1. If the digits are selected at random, what is the probability of getting a code with exactly two 1 s? 6 = Example: What is the probability of rolling a 5 and spinning a 3 on the spinner shown? 1 24 Probability of Dependent Events P(A and B) = P(A) P(B after A) Example: Seven different books are on a shelf in the classroom. If Jewel chooses a book from the shelf to read, and then Cheryl chooses a book from the ones that remain, what is the probability of them choosing Book 1 and Book 2? P(Book 1) = 1 7. The P(Book 2) = 1 6. P(Book 1, then Book 2) 1 1 = Example: A bag contains 5 red marbles and 5 black marbles. What is the probability of drawing a black marble and then a red marble without replacing the first marble before drawing the second marble? P(B) = 5 = 1, then P(R) = , so P(B, then R) = = Example: A troop of 9 girls must elect a leader, an assistant leader, and a treasurer. No one can hold more than one position at a time. What is the probability that the leader is girl #7, the assistant leader is girl #1 and the treasurer is girl #4? P(#7) = 1 9, P(#1) = 1 8, P(#4) = , so P(#7, then #1, then#4) = = Example: Pablo has 7 coins in his pocket. He has 3 nickels, 2 dimes, 1 quarter and 1 half dollar. Pablo removes one coin from his pocket and then removes a second coin without replacing the first. What is the probability that both coins 1 will be nickels? 7 Example: A class of 30 students, 16 girls and 14 boys, must elect 3 different students to the offices of class president, vice president and a secretary. What is the 13 probability that all three positions will be won by the boys? 145 Example: A student chooses a fiction book at random from a list of books and then chooses a western from those remaining. Identify whether the events are independent or dependent. Explain your answer. Math 7 Notes Unit 5: Probability Page 17 of 36

18 Example: A student draws a red marble from a bag of marbles and spins an even number on a spinner. Identify whether the events are independent or dependent. Explain your answer. Example: Harry and Jason decide to play a game of Monopoly even though they have misplaced several game pieces and only can find the dog, a race car, the thimble, a top hat and an iron. Harry chooses his game piece at the start of the game, then Jason chooses his Monopoly game piece, out of the remaining four pieces. Tell whether the event is dependent or independent. Then find the probability of Harry choosing the iron and Jason choosing the dog. Dependent, 1 20 Example: Find the probability of picking a black checker from a bag of 9 black checkers and 6 red checkers, replacing it, and then drawing another black checker. Tell whether the event is dependent or independent. Independent, 9 25 Example: Find the probability of picking a black checker from a bag of 9 black checkers and 6 red checkers, not replacing it, and then drawing another black checker. Tell whether the event is dependent or independent. Dependent, Example: Liang spins each spinner one time. What is the probability that the first spinner will land on an odd number and the second spinner will land on a vowel? Express your answer as a fraction in simplest form. A B. 1 4 C. 1 3 D. 1 6 Example: Two letters are chosen at random from the word ARITHMETIC. Find the probability of choosing one letter that is a vowel, and then choosing one consonant from the letters that remain. Math 7 Notes Unit 5: Probability Page 18 of 36

19 Example: Spinner A is spun and then Spinner B is spun. What is the probability of landing on 1 both times? A B. 5 4 C D Example: Christy rolls a number cube and then chooses a card from a set numbered 1 through 9. What is the probability that she will roll and even number and choose an odd card? A B C. 1 2 D. 5 9 Example: Two number cubes are rolled and added. What is the probability that the sum is equal to 9? A B C D. 1 9 Example: Greg spins the spinner twice. Find the probability that the spinner will land on an even number both times. Express your answers as a fraction in simplest form. A. 1 8 B C. 1 4 D. 1 2 Math 7 Notes Unit 5: Probability Page 19 of 36

20 NVACS 7. SP.C.8b - Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language, (e.g., rolling double sixes ), identify the outcomes in the sample space which compose the event. Example: An experiment consists of rolling a number cube and spinning a spinner (shown at right). Show the sample space of all likely outcomes. Then find the probability of getting a 5 on each. Express your answer as a fraction in simplest form. Answer: or Spinner ,1 1,2 1,3 1,4 1,5 1,6 1,7 1,8 2 2,1 2,2 2,3 2,4 2,5 2,6 2,7 2,8 3 3,1 3,2 3,3 3,4 3,5 3,6 3,7 3,8 4 4,1 4,2 4,3 4,4 4,5 4,6 4,6 4,8 5 5,1 5,2 5,3 5,4 5,5 5,6 5,7 5,8 6 6,1 6,2 6,3 6,4 6,5 6,6 6,7 6, Example: An experiment consists of rolling two fair number cubes. Show the sample space of all equally likely outcomes. Then find the probability of rolling double sixes. Express your answer as a fraction in simplest form. Answer: Math 7 Notes Unit 5: Probability Page 20 of 36

21 NVACS 7. SP.C.8c - Design and use a simulation to generate frequencies for compound events. Let s say we want to design a random number simulation with 50 trials to find the probability of flipping a coin to get heads. Using a TI-84+ graphing calculator, a simulation can be generated easily. Under the MATH key, scroll right to PRB and choose #5 (:random generator integer). Your screen should like this: Press enter. For this problem I choose to use the numbers 0 and 1 to represent heads = 0 and tails=1. I need 50 trials so I enter 0, 1, 50 and my screen will now look like this. Press enter As quick as a flash the calculator has generated 50 trials. Remember to scroll to the right to see all the results. I will create a table here for you to see my results. Trial # # outcome Trial # # outcome 1 1 tails 26 1 tails 2 1 tails 27 0 heads 3 0 heads 28 0 heads Tallying the outcomes I get: 4 1 tails 29 0 heads Heads Tails 5 0 heads 30 1 tails 6 1 tails 31 1 tails 7 0 heads 32 1 tails 8 1 tails 33 1 tails 9 0 heads 34 0 heads tails 35 1 tails 11 0 heads 36 0 heads 12 1 tails 37 0 heads So 27 times out of my 50 flips, I 13 0 heads 38 1 tails 14 0 heads 39 0 heads would get heads. That means 27 54% 50 = heads 40 1 tails 16 1 tails 41 0 heads Is my answer reasonable? Yes I would 17 0 heads 42 1 tails agree it is reasonable. Math 7 Notes Unit 5: Probability Page 21 of 36

22 18 1 tails 43 0 heads 19 0 heads 44 0 heads 20 0 heads 45 1 tails 21 1 tails 46 0 heads 22 0 heads 47 1 tails 23 0 heads 48 0 heads 24 1 tails 49 0 heads 25 1 tails 50 0 heads Example: A full deck of 52 cards contains 26 red cards and 26 black cards. There are 4 suits of cards and 13 cards in each suit. 13 hearts (red) 13 diamonds (red) 13 clubs (black) 13 spades (black) Part A: Design a random-number simulation with 25 trials to find the probability of drawing at least 1 spade out of three draws from the deck (replacing each card before the next draw). Part B: Perform the simulation and find the probability. Part C: Is the result what you expected? Explain your answer. So on my TI-84+ calculator we would type in randint(1, 52, 3) and run it 1 and 52 for the numbers I want randomly generated 3 numbers per trial 25 times for the 25 trials. Math 7 Notes Unit 5: Probability Page 22 of 36

23 Example: A high school band sells pizza as a fund raiser. The results of their last year s sales are shown. Part A: Based on the trends from last year s sales, design a randon-number simulation with 25 trials to generate frequencies for the first 5 pizzas sold. Part B: Perform the simulation, and record the results. Use the results for both parts C and D. Part C: Find the probability that at least 3 of the first 5 sales are pepperoni pizzas. Part D: Find the probability that no more than 2 of the first 5 sales are cheese pizzas. Part E: Describe how your results compare to your expectations. Math 7 Notes Unit 5: Probability Page 23 of 36

24 Looking for more examples and/or readymade worksheets on this standards? Click on the link below. Math 7 Notes Unit 5: Probability Page 24 of 36

25 Sample Questions 1. Abby can create an outfit by wearing one skirt and one top. The skirts and tops she can choose from are described in the lists below. Skirt: black, gray Top: striped, solid, dotted, flowered Abby does not like to wear the two outfits described below. Black skirt and solid top Gray skirt and flowered top Which tree diagram shows all the possible combinations of one skirt and one top that are remaining for Abby to use to create an outfit she does like to wear? A. black gray striped dotted flowered striped solid dotted C. black striped gray solid dotted flowered B. black striped solid dotted D. black striped solid dotted flowered gray solid striped flowered gray striped solid dotted flowered Math 7 Notes Unit 5: Probability Page 25 of 36

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27 2013 SBAC Questions 7.SP.C.5& 7.SP.C.7 DOK: 3 Difficulty: Medium Question Type: ER Extended Response Carl and Beneta are playing a game using this spinner. Carl will win the game on his next spin if the arrow lands on a section labeled 6, 7, or 8. Carl claims it is likely, but not certain, that he will win the game on his next spin. Explain why Carl s claim is not correct. Beneta will win the game on her next spin if the result of the spin satisfies event X. Beneta claims it is likely, but not certain, that she will win the game on her next spin. Describe an event X for which Beneta s claim is correct. Math 7 Notes Unit 5: Probability Page 27 of 36

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