15-4 Compound Probability TEKS FOCUS TEKS (1)(E) Apply independence in contextual problems. TEKS (1)(B) Use a problemsolving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution. Additional TEKS (1)(A), (1)(C) VOCABULARY Compound event an event that consists of two or more events linked by the word and or the word or Dependent events Two events are dependent if the occurrence of one event affects the probability of the other event. Independent events Two events are independent if the occurrence of one event does not affect the probability of the other event. Mutually exclusive events events that cannot happen at the same time Overlapping events events that have at least one outcome in common Formulate create with careful effort and purpose. You can formulate a plan or strategy to solve a problem. Reasonableness the quality of being within the realm of common sense or sound reasoning. The reasonableness of a solution is whether or not the solution makes sense Strategy a plan or method for solving a problem ESSENTIAL UNDERSTANDING You can find the probability of a compound event by using the probability of each part of the compound event. Key Concept Probability of Independent Events If A and B are independent events, then P (A and B) = P (A) # P (B). Key Concept Probability of Mutually Exclusive Events If A and B are mutually exclusive events, then P(A and B) = 0, and P (A or B) = P (A) + P (B). Key Concept Probability of Overlapping Events If A and B are overlapping events, then P (A or B) = P (A) + P(B) - P (A and B). 628 Lesson 15-4 Compound Probability
Problem 1 Identifying Independent and Dependent Events Are the outcomes of each trial independent or dependent events? How can you tell that two events are independent? Two events are independent if one does not affect the other. A Choose a number tile from 12 tiles. Then spin a spinner. The choice of number tile does not affect the spinner result. The events are independent. B Pick one card from a set of 15 sequentially numbered cards. Then, without replacing the card, pick another card. The first card chosen affects the possible outcomes of the second pick, so the events are dependent. Problem 2 Finding the Probability of Independent Events A desk drawer contains red pens, blue pens, black pens, silver paper clips, and white paper clips. If you select a pen and a paper clip from the drawer without looking, what is the probability that you select a blue pen and a white paper clip? 24 silver paper clips 16 white paper clips Why are the events independent? Selecting a blue pen has no affect on selecting a white paper clip. Step 1 Let A = selecting a blue pen. Find the probability of A. Step 2 P (A) = 14 = 7 6 blue pens out of 14 pens Let B = selecting a white paper clip. Find the probability of B. Step P (B) = 40 = 25 Find P (A and B). 6 16 P (A and B) = P (A) 16 white paper clips out of 40 clips Use the formula for the probability of independent events. # P (B) = 7 # 25 = 56 0.171, or 17.1% The probability that you select a blue pen and a white paper clip is about 17.1%. PearsonTEXAS.com 629
Problem TEKS Process Standard (1)(A) Finding the Probability of Mutually Exclusive Events Is there a way to simplify this problem? You can model the probabilities with a simpler problem. Suppose there are 100 athletes. In the model 10 athletes will play volleyball, and 24 will be on the swim team. Athletics Student athletes at a local high school may participate in only one sport each season. What is the probability that a randomly selected student athlete plays volleyball or is on the swim team? Because athletes participate in only one sport each season, the events are mutually exclusive. Use the formula P (A or B) = P (A) + P (B). P (volleyball or swim team) = P (volleyball) + P (swim team) = 10% + 24% = 4% Substitute and simplify. The probability of a student athlete either playing volleyball or being on the swim team is 4%. KEY: Fall Season Sports 24% 28% 18% 20% 10% Basketball Volleyball Football Other Swimming Problem 4 TEKS Process Standard (1)(B) Finding Probabilities of Overlapping Events What is the probability of rolling either an even number or a multiple of when rolling a standard number cube? Why do you need to subtract the overlapping probability? If the overlapping probability is not subtracted, it is counted twice. This would introduce an error. You are rolling a standard number cube. The events are overlapping events because 6 is both even and a multiple of. You need the probability of rolling an even number and the probability of rolling a multiple of. Find the probabilities and use the formula for probabilities of overlapping events. P (even or multiple of ) = P (even) + P (multiple of ) - P (even and multiple of ) = 6 + 2 6-1 6 = 4 6, or 2 The probability of rolling an even number or a multiple of is 2. 60 Lesson 15-4 Compound Probability
ONLINE H O M E W O R K PRACTICE and APPLICATION EXERCISES Scan page for a Virtual Nerd tutorial video. Determine whether the outcomes of the two actions are independent or dependent events. For additional support when completing your homework, go to PearsonTEXAS.com. 1. You toss a coin and roll a number cube. 2. You draw a marble from a bag without looking. You do not replace it. You draw another marble from the bag.. Choose a card at random from a standard deck of cards and replace it. Then choose another card. 4. Ask a student s age and ask what year the student expects to graduate. You spin the spinner at the right and, without looking, you choose a tile from a set of tiles numbered from 1 to 10. Find each probability. 5. P (spinner lands on 2 and you choose a ) 6. P (spinner lands on an odd number and you choose an even number) 7. P (spinner lands on a number less than 4 and you choose a 9 or 10) A bag contains blue chips, 6 black chips, 2 green chips, and 4 red chips. Use this information to find each probability if a chip is selected at random. 8. P (blue chip or black chip) 9. P (green chip or red chip) 10. P (green chip or black chip) 11. P (blue, black, or red chip) A set of cards contains four suits (red, blue, green, and yellow). In each suit there are cards numbered from 1 to 10. Calculate the following probabilities for one card selected at random. 12. P (blue card or card numbered 10) 1. P (green or yellow card, or card numbered 1) 14. P (red card or card greater than 5) 15. P (red or blue card, or card less than 6) 16. Apply Mathematics (1)(A) In a litter of 8 kittens, there are 2 brown females, 1 brown male, spotted females, and 2 spotted males. If a kitten is selected at random, what is the probability that the kitten will be female or brown? 17. Analyze Mathematical Relationships (1)(F) Suppose you are taking a test and there are three multiple-choice questions that you do not know the answers to. Each has four answer choices. Rather than leave the answers blank, you decide to guess. What is the probability that you answer all three questions correctly? Explain how you know. 1 2 4 PearsonTEXAS.com 61
18. Use a Problem-Solving Model (1)(B) In a math class, 75% of the students have visited the ocean and 50% have visited the mountains on vacation before. If 45% of the students have visited the ocean and the mountains on vacation before, what is the probability that a randomly selected student has visited the ocean or the mountains? 19. What is the probability that a standard number cube rolled three times will roll first even, then odd, and then even? 20. Explain Mathematical Ideas (1)(G) Describe the difference between mutually exclusive and overlapping events. Give examples of each. 21. When you draw a marble out of a bag and then draw another without replacing the first, the probability of the second event is different from the probability of the first. a. What is the probability of drawing a red marble out of a bag containing red and 7 blue marbles? b. What is the probability of drawing a second red marble if a red marble is drawn the first time and not replaced? c. What is the probability of drawing two red marbles in a row? For each set of probabilities, determine if the events A and B are mutually exclusive. Explain. 22. P (A) = 1 2, P(B) = 1, P(A or B) = 2 2. P (A) = 1 6, P(B) = 8, P(A and B) = 0 24. Connect Mathematical Ideas (1)(F) Are mutually exclusive events dependent or independent? Explain. TEXAS Test Practice 25. Which of the following statements is NOT true? A. The side lengths of an isosceles right triangle can be all whole numbers. B. The side lengths of a right triangle can form a Pythagorean triple. C. The side lengths of an equilateral triangle can be all whole numbers. D. The angle measures of an equilateral triangle can be all whole numbers. 26. You roll a standard number cube and then spin the spinner shown at the right. What is the probability that you will roll a 5 and spin a? 27. An arc of a circle measures 90 and is 10 cm long. How long is the circle s diameter? 6 1 2 5 4 62 Lesson 15-4 Compound Probability