bell work 1.mr. chou is redecorating his office. he has a choice of 4 colors of paint, 3 kinds of...
TRANSCRIPT
Bell Work1. 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
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 ChoiceSecond Choice
3 3P(red and red) =
10 109
100x
710
red
blue
red
blue
310 3 7
P(red and blue) =10 10
21100
x
7 3P(blue and red) =
10 1021
100x
7 7P(blue and blue) =
10 1049100
x
310
710
310
710
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
59
black
green
black
green
49 4 5
P(black and green) =9 9
2081
x
5 5P(green and green) =
9 92581
x
49
59
49
59
4 4P(black and black) =
9 91681
x
5 4P(green and black) =
9 92081
x
Independent Events
Q2 Coins
head
tail
First Coin Second Coin
12
head
tail
head
tail
12 1 1
P(head and tail) 2
1=
2 4x
1 1P(tail and tail)
21
=2 4
x
12
12
12
12
1 1P(head and head)
21
=2 4
x
1 1P(tail and head)
21
=2 4
x
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
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 TennisPeter Win
Peter Win
0.40.7
0.6
0.3
0.3
0.7
0.4 x 0.3 = 0.12
0.4 x 0.7 = 0.28
0.6 x 0.3 = 0.18
0.6 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 Drawred
blue
blue
yellowred
blue
yellowred
blue
yellow
3 Independent Events
520
420
1120
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
520
red
420
420
520
blue
1120
blue
yellowred
blue
yellowred
blue
yellow
1120
420
520
1120
420
520
1120
3 Independent Events
1. P(blue, blue)
2. P(yellow then blue)
3. P(red and yellow)
1120 x
1120 = 121/400
Or 30.25%
520
x1120
= 55/400 = 11/80 or 13.75%
= 40/400 = 1/10Or 10%
420
x5
20 +4
20x
520
Probability (Tree Diagrams)
red
First Draw Second Draw
710
red310
blue
2 Independent Events. 3 Selections
red
red
red
red
blue
blue
blue
blue
red
blue
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
710
red310
blue
2 Independent Events. 3 Selections
red
red
red
red
blue
blue
blue
blue
310
310
710310
710
710
310
710
310
710 3
10
710
red
blue
blue
Third Draw
GUIDED PRACTICEYou Try:1. You toss a coin twice. Find the probability of getting two heads.
P(head and head) = P(head) P(head) = 14
or 25%12
12
=
(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)
= 80729
or 25%
49
59
=49
Practice:Worksheet on
Independent Events