Math 7 Notes - Unit 7B (Chapter 11) Probability

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1 Math 7 Notes - Unit 7B (Chapter 11) Probability Probability Syllabus Objective: (7.2)The student will determine the theoretical probability of an event. Syllabus Objective: (7.4)The student will compare theoretical and experimental probabilities of an event. Syllabus Objective: (7.5)The student will represent the probability of an event as a number between 0 and 1. 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. Both probability and odds are ways to tell how likely it is that an event will or will not happen. Note to CCSD teachers: The textbook uses words or phrases to represent the probability of an event from impossible to certain. The CRT requires number values to represent probability (from 0 to 1). Probability is the measure of how likely an event is to occur. They are written as fractions or decimals from 0 to 1. Probability may be written 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 2 or 50% has the same chance of happening as not happening. Example: How likely is it that a coin tossed will come up heads? This means that there is as likely a chance of heads as not heads. In other words, a probability of 1 or 0.5 or 50%. 2 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 Math 7, Unit 7B (Ch 11): Probability Holt: Chapter 11 Page 1 of 9

2 Outcomes are the possible results of an experiment. Example: Tossing a coin; the possible outcomes (results) are a head or a tail. 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. Mathematically, we write: probability = number of favorable outcomes number of possible outcomes or 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? P( yellow marbles ) = # of yellow marbles total number of marbles 2 1 P = or 6 3 The probability found in the above example is an example of theoretical probability. Experimental probability is 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 ) = 30 Math 7, Unit 7B (Ch 11): Probability Holt: Chapter 11 Page 2 of 9

3 Odds Note to CCSD teachers: odds is not taught in the textbook; however, this concept is included in the CCSD benchmarks and Nevada Standards and will be tested on the CRT. Also, teachers should be aware that one can convert from odds to probability, and vice versa. For example, if the odds of winning a game is 2 3, then the probability of winning is 2. If the 5 probability of rain is 7 10, then the odds of rain is 7 3. Syllabus Objective: (7.3) The student will determine the odds of an event. Odds: the ratio of favorable outcomes to the number of unfavorable outcomes, when all outcomes are equally likely. Odds in favor = Odds against = Number of favorable outcomes Number of unfavorable outcomes Number of unfavorable outcomes Number of favorable outcomes Example: Suppose you pick a marble from a hat that contains three red, two yellow and one blue marble. What are the odds (in favor) you draw a yellow marble? 2 of yellow marbles 2 1 P = = = or 1:2 # of marbles not yellow 4 2 Example: If the probability of an event is 4, find the odds = 5, so the odds are. 5 Example: At a carnival ring toss game, an average of 3 people in 10 win a prize. Give the odds against winning the prize. number of favorable outcomes number of unfavorable outcomes = 7 3 Math 7, Unit 7B (Ch 11): Probability Holt: Chapter 11 Page 3 of 9

4 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? 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. A tree diagram makes it easier to see (count) the number of possible outcomes for experiments when the numbers are small and there are multiple events. To draw a tree diagram, you: 1) begin with a point; then you draw a segment for each outcome in the first event. 2) draw segments 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 segments and label H and T. H T 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. Math 7, Unit 7B (Ch 11): Probability Holt: Chapter 11 Page 4 of 9

5 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? 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. Math 7, Unit 7B (Ch 11): Probability Holt: Chapter 11 Page 5 of 9

6 Fundamental Counting Principle Tree diagrams are useful to get a picture of what is occurring, but with a large number of events, the tree can get out of hand in a hurry. A quick way to determine the number of possible outcomes in a tree diagram is to multiply the number of outcomes in each event. With the first example 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 last 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 and 2 3= 6. 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 = 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. If the occurrence of one event does have an effect on the probability that the second event will occur, the events are dependent. Math 7, Unit 7B (Ch 11): Probability Holt: Chapter 11 Page 6 of 9

7 Examples of independent and dependent events Example: A student rolls a number cube, then rolls a second time and records each result. These events are independent since the outcome of rolling one number cube does not affect the outcome of rolling the number cube the second time. Example: One student chooses a book off a library shelf. A second student then choses a different book off the same shelf from the remaining books. These events are dependent since the students must choose different books and the second student has fewer books to choose from. Example: A student rolls a number cube and then flips a coin. These events are independent since the rolling of a number cube does not affect the outcome of flipping a coin. Example: the first. A student draws an item from a bag, then draws a second item without replacing These events are dependent since the outcome of the first draw affects the outcome of the second draw (remember the second draw contains one less item than the first draw). Example: A student draws an item from a bag, replaces it, then draws a second item. These events are independent since the outcome of the first draw does not affect the outcome of the second draw 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 shirt, P(T) = 1 3, for the shorts P(S) = ; So P(T, S) = or Math 7, Unit 7B (Ch 11): Probability Holt: Chapter 11 Page 7 of 9

8 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 = Remember when Cheryl chooses a book there is one less book on the shelf. So 7 1 = 6 OnCore Examples 1. In a hat, you have index cards with the numbers 1 through 10 written on them. You pick one card at random, Order the events from least likely to happen to most likely to happen. You pick a number greater than 0. You pick an even number. You pick a number that is at least 2. You pick a number that is at most Determine whether each event below is impossible, unlikely, as likely as not, likely, or certain. Then tell whether the probability is 0, close to 0, 1, close to 1 or 1. 2 A. The probability of rolling a 5 on a number cube is 1. What is the probability of 6 not rolling a 5? B. Picking a number less than 15 from a jar with papers labeled from 1 to 12. C. Picking a number that is divisible by 5 from a jar with papers labeled from 1 to Describe an event that has a probability of 0% and an event that has a probability of 100%. Math 7, Unit 7B (Ch 11): Probability Holt: Chapter 11 Page 8 of 9

9 Applicable CCSS Example (Although technically it has not rolled out yet but a similar standard in NSS is still in place) Standard: 7.SP.5 DOK: 3 Difficulty: Medium Type: 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, Unit 7B (Ch 11): Probability Holt: Chapter 11 Page 9 of 9