. Independent Events Lesson Objectives Understand independent events. Use the multiplication rule and the addition rule of probability to solve problems with independent events. Vocabulary independent events multiplication rule of probability addition rule of probability Understand Independent Events. Suppose you are playing a game. You have a spinner with two congruent sections and some color cards as shown below. Your goal is to randomly spin a and draw a red card. B Y The event of spinning a and the event of drawing a red card are considered independent events. Two events are independent if the occurrence of one event does not affect the probability of the other event. When you spin the spinner, regardless of the result, it will not affect the probability of drawing a blue, yellow, or red card. Caution Independent events are not the same as mutually exclusive events. Mutually exclusive events cannot occur at the same time. But independent events refer to whether the occurrence of an event affects the probability of the other event. Chapter Probability
Use the Multiplication ule of Probability to Solve Problems with Independent Events. Consider the spinner and the color cards again. You can draw a tree diagram to represent the independent events that form the compound event and their corresponding probabilities. Spinner Color Card Outcome B (, B) (, ) Y (, Y) B (, B) (, ) B represents blue Y (, Y) represents red Y represents yellow Because both events are independent, the outcome on the spinner does not affect the probability of drawing a color card. The area of the spinner is equally divided into sections, so the probability of spinning one of the two numbers is. Because there is an equal chance of drawing each color card, the probability of choosing one of the three colors cards is. The probabilities are labeled on the branches of the tree diagram. You can see that there are a total of equally likely outcomes, and spinning a and drawing a red card is one of those equally likely outcomes. So, you can write the probability of spinning a and drawing a red card as follows: P(, ) 5 You can also use the multiplication rule of probability to find the probability of spinning a and drawing a red card. P(, ) 5 P() P() 5 Multiply P() and P(). 5 Simplify. Continue on next page Lesson. Independent Events 7
In general, for two independent events A and B, the multiplication rule of probability states that: P(A and B) 5 P(A) P(B) Suppose the blue card is replaced by another red card as shown. Y A tree diagram can be drawn to represent the possible outcomes as shown. Spinner Color Card Outcome (, ) (, ) Y (, Y) (, ) (, ) Y (, Y) represents red Y represents yellow You can see that there is a total of outcomes. To find the probability of spinning a and drawing a red card, you can see that out of outcomes are favorable. So, you can write the probability of spinning a and drawing a red card as: P(, ) 5 5 8 Chapter Probability
The tree diagram constructed previously for this compound event with two red cards can be simplified by combining the identical outcomes (, ) and (, ) together as shown below. Spinner Color Card Outcome (, ) Y (, Y) (, ) Y (, Y) represents red Y represents yellow You can clearly see that the probability of drawing a red card is greater than in the last example when drawing each color was equally likely. The simple event of drawing a color card has become a biased event, because the probability of drawing a red card is not the same as the probability of drawing a yellow card. Using the multiplication rule to find the probability of spinning a and drawing a red card: P(, ) 5 P() P() 5 Multiply P() and P(). 5 Simplify. Caution For compound events involving biased outcomes, the probability of an outcome is not necessarily equal to Number of different outcomes because the outcomes are not equally likely. Lesson. Independent Events 9
Example Solve probability problems involving two independent events. A game is played with a fair coin and a fair six-sided number die. To win the game, you need to randomly obtain heads on a fair coin and a on a fair number die. a) Draw a tree diagram to represent this compound event. Solution Coin Number Die Outcome (H, ) (H, ) (H, ) The events are independent since throwing a coin and a number die do not affect the results of each other. H (H, ) 5 (H, 5) (H, ) (T, ) (T, ) (T, ) T (T, ) 5 (T, 5) H represents heads (T, ) T represents tails 0 Chapter Probability
b) Use the multiplication rule of probability to find the probability of winning the game in one try. Solution P(winning the game) 5 P(H, ) 5 P(H) P() 5 Think Math If the coin and number die are biased, could the chance of winning the game be the same Explain. 5 The probability of winning the game in one try is. uided Practice Solve. Show your work. A game is played with a bag of color tokens and a bag of letter tiles. The tokens consist of green tokens, yellow token, and red tokens. The letter tiles consist of tiles of letter A and tiles of letter B. To win the game, you need to get a yellow token and a tile of letter B from each bag. a) Copy and complete the tree diagram. Token Letter Tile Outcome A (, ) Y B A (, ) (, ) B A B (, ) (, ) (, ) represents green Y represents yellow represents red A represents letter A B represents letter B b) Use the multiplication rule of probability to find the probability of winning the game in one try. P(winning the game) 5 P(Y, B) 5 P(Y) P(B) 5 5 The probability of winning the game in one try is. Lesson. Independent Events
Technology Activity Materials: spreadsheet software SIMULATE ANDOMNESS Work in pairs. Background Two fair six-sided number dice are thrown. Using a spreadsheet, you can generate data to investigate how frequently the outcome of doubles ( and, and,, and ) occurs. STEP Label your spreadsheet as shown. STEP To generate a random integer between and in cell A, enter the formula 5 INT(AND()*) to simulate rolling a die. A random number from to should appear in the cell. STEP To model 00 rolls, select cells A to A0 and choose Fill Down from the Edit menu. STEP epeat STEP and STEP for cells B to B0. Chapter Probability
STEP 5 In cell C, enter the formula 5 A B. Select cells C to C0 and choose Fill Down from the Edit menu. This column serves as a check to see if the random numbers generated in columns A and B are the same. If the numbers are the same, their difference is 0. A zero difference indicates doubles outcome. STEP To see how many times the data show doubles occurring, enter the formula 5 COUNTIF(C:C0,0) in cell D. STEP 7 Find the experimental probability of the occurrence of two number dice showing the same number by dividing the number you get in cell D by the total, 00 rolls. Find the theoretical probability of rolling doubles with fair number dice. Compare this theoretical probability with the experimental probability you obtained in the spreadsheet simulation. Are these two values the same When you use a greater number of simulations, such as 00 instead of 0, the result is more likely to be closer to the theoretical probability. Lesson. Independent Events
Example 7 Solve probability problems involving independent events with replacement. A jar contains 8 green marbles and red marbles. One marble is randomly drawn and the color of the marble is noted. The marble is then put back into the jar and a second marble is randomly drawn. The color of the second marble is also noted. a) Find the probability of first drawing a green marble followed by a red marble. Solution 8 st Draw nd Draw Outcome 8 (, ) (, ) Since the first marble is drawn and replaced, the probability of drawing the second marble remains unchanged. 8 (, ) (, ) represents green represents red P(, ) 5 P() P() 5 8 5 9 The probability of first drawing a green marble followed by a red marble is 9. b) Find the probability of first drawing a red marble followed by a green marble. Solution P(, ) 5 P() P() 5 8 5 9 The probability of first drawing a red marble followed by a green marble is 9. c) Find the probability of drawing two green marbles. Solution P(, ) 5 P() P() 5 8 8 5 9 The probability of drawing two green marbles is 9. Chapter Probability
uided Practice Solve. Show your work. In a bag, there are 9 magenta balls and orange ball. Two balls are randomly drawn, one at a time with replacement. a) Find the probability of drawing two magenta balls. st Draw nd Draw Outcome O (O, O) O 0 M (O, M) 9 0 O (M, O) M M (M, M) O represents orange M represents magenta P(M, M) 5 P(M) P(M) 5 5 The probability of drawing two magenta balls is. b) Find the probability of drawing an orange ball followed by a magenta ball. P(O, M) 5 P(O) P(M) 5 5 The probability of drawing an orange ball followed by a magenta ball is. c) Find the probability of drawing an orange ball both times. P(O, O) 5 P(O) P(O) 5 5 The probability of drawing an orange ball both times is. Lesson. Independent Events 5
Use the Addition ule of Probability to Solve Problems with Independent Events. You have learned how to use the multiplication rule of probability to find the probability of one favorable outcome in a compound event. Now you will learn to use the addition rule of probability to find the probability of more than one favorable outcome in a compound event. A jar contains 8 green marbles and red marbles. One marble is randomly drawn and the color of the marble is noted. The marble is then put back into the jar and a second marble is randomly drawn. The color of the second marble is also noted. st Draw nd Draw Outcome 8 (, ) 8 (, ) 8 (, ) (, ) represents green represents red Suppose you want to find the probability of drawing two marbles of the same color. There are two favorable outcomes, (, ) and (, ), and they are mutually exclusive. P(, ) 5 5 9 P(, ) 5 8 8 5 9 To find the probability of (, ) or (, ), you can find the sum of their probabilities. Using the addition rule of probability: P(same color) 5 P(, ) P(, ) 5 9 9 5 5 9 Math Note Because (, ) and (, ) are mutually exclusive events, you can add the probabilities to find the probability of (, ) or (, ). In general, for two mutually exclusive events A and B, the addition rule of probability states that: P(A or B) 5 P(A) P(B) Chapter Probability
Example 8 Solve probability problems with independent events involving more than one favorable outcome. Alex is taking two tests. The probability of him passing each test is 0.8. a) Find the probability that Alex passes both tests. Solution To draw a tree diagram, first find the probability that Alex fails the test. Let P represent pass and F represent fail. P(F) 5 P(P) 5 0.8 5 0. Math Note ecall that for two complementary events, the sum of their probabilities is. st Test nd Test Outcome (0.8) P (P, P) P (0.8) (0.) F (P, F) (0.) F (0.8) (0.) P F (F, P) (F, F) P represents pass F represents fail P(P, P) 5 P(P) P(P) 5 0.8 0.8 5 0. The probability that Alex passes both tests is 0.. b) Find the probability that he passes exactly one of the tests. Solution Using the addition rule of probability: P((P, F) or (F, P)) 5 P(P, F) P(F, P) 5 P(P) P(F) P(F) P(P) 5 0.8 0. 0. 0.8 5 0. The probability that he passes exactly one of the tests is 0.. Alex passes exactly one of the tests means that he either passes the first test or the second test. So, there are two possible cases. Think Math What is the probability of passing at least one test Show your reasoning. Lesson. Independent Events 7
uided Practice Solve. Show your work. On weekends, Carli either jogs (J) or plays tennis (T) each day, but never both. The probability of her playing tennis is 0.75. a) Find the probability that Carli jogs on both days. Because J and T are complementary, P(J) 5 P(T) 5 5 Saturday Sunday Outcome J (, ) ( ) J ( ) (0.75) T (, ) (0.75) ( ) J (, ) T (0.75) T (, ) J represents jog T represents tennis P(J, J) 5 P(J) P(J) 5 5 The probability that Carli jogs on both days is. b) Find the probability that Carli jogs on exactly one of the days. Using the addition rule of probability: P(J, T) or P(T, J) 5 P(J, T) P(T, J) 5 P(J) P(T) P(T) P(J) 5 5 The probability that Carli jogs on exactly one of the days is. 8 Chapter Probability
Practice. Draw a tree diagram to represent each situation. Tossing a fair coin followed by drawing a marble from a bag of marbles: yellow, green, and blue Drawing two balls randomly with replacement from a bag with green ball and purple ball Drawing a ball randomly from a bag containing red ball and blue ball, followed by tossing a fair six-sided number die Tossing a fair coin twice 5 eading or playing on each day of a weekend On time or tardy for school for two consecutive days Solve. Show your work. 7 Mindy is playing a game that uses the spinner shown below and a fair coin. An outcome of on the spinner and heads on the coin wins the game. a) Draw a tree diagram to represent all possible outcomes and the corresponding probabilities. b) Find the probability of winning the game in one try. c) Find the probability of losing the game in one try. 8 There are blue balls and yellow balls in a bag. A ball is randomly drawn from the bag, and it is replaced before a second ball is randomly drawn. a) Draw a tree diagram to represent all possible outcomes. b) Find the probability that a yellow ball is drawn first, followed by another yellow ball. c) Find the probability that a yellow ball is drawn after a blue ball is drawn first. Lesson. Independent Events 9
9 Jasmine has blue pens and green pens in her pencil case. She randomly selects a pen from her pencil case, and replaces it before she randomly selects again. a) Draw a tree diagram to represent all possible outcomes and the corresponding probabilities. b) Find the probability that she selects blue pens. c) Find the probability that she selects green pens. d) Find the probability that she selects pens of the same color. 0 Henry has fiction books, nonfiction books, and Spanish book on his bookshelf. He randomly selects two books with replacement. a) Draw a tree diagram to represent all possible outcomes and the corresponding probabilities. b) Find the probability that he selects a fiction book twice. c) Find the probability that he first selects a nonfiction book, and then a Spanish book. d) Find the probability that he first selects a fiction book, and then a nonfiction book. Andy tosses a fair six-sided number die twice. What is the probability of tossing an even number on the first toss and a prime number on the second toss The probability that Fiona wakes up before 8 A.M. when she does not set her alarm is. On any two consecutive days that Fiona does not set her alarm, what is the 5 probability of her waking up before 8 A.M. for at least one of the days 50 Chapter Probability
A globe is spinning on a globe stand. The globe s surface is painted 0% yellow, 0% green, and the rest is painted blue. Two times Danny randomly points to a spot on the globe while it spins. The color he points to each time after the spinning stops is recorded. a) What is the probability that he points to the same color on both spins b) What is the probability that he points to yellow at least one time Sally thinks that for two independent events, because the occurrence of one event will not have any impact on the probability of the other event, they are also mutually exclusive. Do you agree with her Explain your reasoning using an example. 5 A game is designed so that a player wins when the game piece lands on letter A. The game piece begins on letter. A fair six-sided number die is tossed. If the number tossed is odd, the game piece moves one step counterclockwise. If the number tossed is even, the game piece moves one step clockwise. a) What is the probability that a player will win after tossing the number die once b) What is the probability that a player will win after tossing the number die twice A M E E A M Lesson. Independent Events 5