Bell Work 1.Mr. Chou is redecorating his office. He has a choice of 4 colors of paint, 3 kinds of curtains, and 2 colors of carpet. How many different combinations of paint, curtains, and carpets can he use? 2.You have 6 posters to hang up on the wall. How many different ways can you hang the posters? 4 x 3 x 2 = 24 combinations 6 x 5 x 4 x 3 x 2 x 1 = 720 different ways

Independent and Dependent Events

Tell whether the events are independent or dependent. You randomly draw a number from a bag. Then you randomly draw a second number without putting the first number back. b. You roll a number cube. Then you roll the number cube again. a. The result of the first roll does not affect the result of the second roll, so the events are independent. There is one fewer number in the bag for the second draw, so the events are dependent.

You Try In Exercises 1 and 2, tell whether the events are independent or dependent. Explain your reasoning. 1. You toss a coin. Then you roll a number cube. You randomly choose 1 of 10 marbles. Then you randomly choose one of the remaining 9 marbles. 2. The coins toss does not affect the roll of a dice, so the events are independent. There is one fewer number in the bag for the second draw, so the events are dependent.

Independent red blue First Choice Second Choice red blue red blue Tree diagrams can be used to help solve problems involving both dependent and independent events. The following situation can be represented by a tree diagram. Peter has ten cubes in a bag. Three of the cubes are red and 7 are blue. He removes a cube at random from the bag and notes the color before replacing it. He then chooses a second cube at random. Record the information in a tree diagram and find the probability of drawing each combination in that order Probability (Tree Diagrams) Independent Events

Rebecca has nine beads in a bag. Four of the beads are black and the rest are green. She removes a bead at random from the bag and notes the color before replacing it. She then chooses a second bead. (a) Draw a tree diagram showing all possible outcomes. (b) Calculate the probability that Rebecca chooses: (i) 2 green beads (ii) A black followed by a green bead. You Try: black green First Choice Second Choice black green black green Independent Events

Q2 Coins head tail First Coin Second Coin head tail head tail Peter tosses two coins. (a) Draw a tree diagram to show all possible outcomes. (b) Use your tree diagram to find the probability of getting (i) 2 Heads (ii) A head or a tail in any order. Probability (Tree Diagrams) P(2 heads) = ¼ P(head and a tail or a tail and a head) = ½ Independent Events

Q3 Sports Becky Win Peter and Becky run a race and play a tennis match. The probability that Peter wins the race is 0.4. The probability that Becky wins the tennis is 0.7. (a) Complete the tree diagram below. (b) Use your tree diagram to calculate (i) the probability that Peter wins both events. (ii) The probability that Becky loses the race but wins at tennis. Probability (Tree Diagrams) Peter Win Becky Win Race Tennis Peter Win x 0.3 = x 0.7 = x 0.3 = x 0.7 = 0.42 P(Win and Win) for Peter = 0.12 P(Lose and Win) for Becky = 0.28 Independent Events

3 Ind/Blank Probability (Tree Diagrams) red yellow First Draw Second Draw red blue yellow red blue yellow red blue yellow 3 Independent Events You choose a colored cube and then replace it. Finish the tree diagram for the second draw.

Probability (Tree Diagrams) red yellow First Draw Second Draw red blue yellow red blue yellow red blue yellow 3 Independent Events 1. P(blue, blue) 2. P(yellow then blue) 3. P(red and yellow) x= 121/400 Or 30.25% x = 55/400 = 11/80 or 13.75% = 40/400 = 1/10 Or 10% x+x

Probability (Tree Diagrams) red First Draw Second Draw red blue 2 Independent Events. 3 Selections red blue red blue Third Draw You choose a colored chip and then replace it. Finish the tree diagram for the second and third draw.

3 Ind/3 Select Probability (Tree Diagrams) red First Draw Second Draw red blue 2 Independent Events. 3 Selections red blue red blue Third Draw

GUIDED PRACTICE You Try: 1. You toss a coin twice. Find the probability of getting two heads. P( head and head ) = P( head ) P( head ) = 1 4 or 25% = (The tosses are independent events, because the outcome of a toss does not affect the probability of the next toss) 2. You draw from a bag of marbles that has 4 red marbles and 5 black marbles and replace it each time. Find the probability of drawing a red, then a black then a red. P( red, black, red) = P( red ) P( black) P ( red) = or 25% = 4 9

Practice: Worksheet on Independent Events