bell work put your homework on your desk—we will discuss it at 8:30 make sure you have the...
TRANSCRIPT
Bell Work
• Put your homework on your desk—we will discuss it at 8:30
• Make sure you have the following markers: red, blue, green and black
• Make sure you have two pieces of graph paper
• Give Durfee eye contact when you are ready to begin
Independent
red
blue
First Choice Second Choice3 3
P(red and red) =10 10
9100
x
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 coloured 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 colour before replacing it. He then chooses a second cube at random. Record the information in a tree diagram.
Probability (Tree Diagrams)
Characteristics
Probability (Tree Diagrams)
The probabilities
for each event are shown
along the arm of each
branch and they sum to 1.
red
blue
First Choice Second Choice3 3
P(red and red) =10 10
9100
x
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
Ends of first and second level
branches show the different outcomes.
Probabilities are multiplied along
each arm.
Characteristics of a tree diagram
Q1 beads
Question 1 Rebecca has nine coloured 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 colour 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.
Probability (Tree Diagrams)
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
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
Question 2 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
Q3 Sports
Becky Win
Becky Win
Question 3 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
Dependent
red
blue
First Choice Second Choice3 2
P(red and red) =10 9
690
x
710
red
blue
red
blue
310 3 7
P(red and blue) =10 9
2190
x
7 3P(blue and red) =
10 92190
x
7 6P(blue and blue) =
10 94290
x
29
79
39
69
The following situation can be represented by a tree diagram.
Peter has ten coloured cubes in a bag. Three of the cubes are red and seven are blue. He removes a cube at random from the bag and notes the colour but does not replace it. He then chooses a second cube at random. Record the information in a tree diagram.
Probability (Tree Diagrams)
Dependent Events
Q4 beads
Question 4 Rebecca has nine coloured 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 does not replace it. She then chooses a second bead. (a) Draw a tree diagram showing all possible outcome (b) Calculate the probability that Rebecca chooses: (i) 2 green beads (ii) A black followed by a green bead.
Probability (Tree Diagrams)
Dependent Events
black
green
First Choice Second Choice
59
black
green
black
green
49 4 5
P(black and green) =9 8
2072
x
5 4P(green and green) =
9 82072
x
38
58
48
48
4 3P(black and black) =
9 81272
x
5 4P(green and black) =
9 82072
x
Q5 Chocolates
Question 5 Lucy has a box of 30 chocolates. 18 are milk chocolate and the rest are dark chocolate. She takes a chocolate at random from the box and eats it. She then chooses a second. (a) Draw a tree diagram to show all the possible outcomes. (b) Calculate the probability that Lucy chooses: (i) 2 milk chocolates. (ii) A dark chocolate followed by a milk chocolate.
Probability (Tree Diagrams)
Dependent Events
Milk
Dark
First Pick Second Pick
1230
Milk
Dark
Milk
Dark
1830
18 17P(milk and milk) =
30687030 29
x 1729
1229
1829
1129
18 12P(milk and dark) =
21687030 29
x
12 18P(dark and milk) =
21687030 29
x
12 11P(dark and dark) =
13287030 29
x
3 Ind
Probability (Tree Diagrams)
red
yellow
First Choice Second Choice
520
red
420
420
520
blue
1120
blue
yellowred
blue
yellowred
blue
yellow
1120
420
520
1120
420
520
1120
3 Independent Events
3 Ind/Blank
Probability (Tree Diagrams)
red
yellow
First Choice Second Choicered
blue
blue
yellowred
blue
yellowred
blue
yellow
3 Independent Events
520
420
1120
3 Ind/Blank/2
Probability (Tree Diagrams)
First Choice Second Choice
3 Independent Events
3 Dep
Probability (Tree Diagrams)
red
yellow
First Choice Second Choice
520
red
420
319
519
blue
1120
blue
yellowred
blue
yellowred
blue
yellow
1119
419
519
1019
419
419
1119
3 Dependent Events
3 Dep/Blank
Probability (Tree Diagrams)
red
yellow
First Choice Second Choicered
blue
blue
yellowred
blue
yellowred
blue
yellow
3 Dependent Events
520
420
1120
3 Dep/Blank/23 Dep/Blank
Probability (Tree Diagrams)
First Choice Second Choice
3 Dependent Events
3 Ind/3 Select
Probability (Tree Diagrams)
red
First Choice Second Choice
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 Choice
3 Ind/3 Select/Blank
Probability (Tree Diagrams)
red
First Choice Second Choice
710
red310
blue
2 Independent Events. 3 Selections
red
red
red
red
blue
blue
blue
blue
red
blue
blue
Third Choice
3 Ind/3 Select/Blank2
Probability (Tree Diagrams)
First Choice Second Choice
2 Independent Events. 3 Selections
Third Choice
3 Dep/3 Select
Probability (Tree Diagrams)
red
First Choice Second Choice
710
red310
blue
2 Dependent Events. 3 Selections
red
red
red
red
blue
blue
blue
blue
29
18
79
39
69
78
28
68
28
68 3
8
58
red
blue
blue
Third Choice
3 Dep/3 Select/Blank3 Dep/3 Select
Probability (Tree Diagrams)
red
First Choice Second Choice
710
red310
blue
2 Dependent Events. 3 Selections
red
red
red
red
blue
blue
blue
blue
red
blue
blue
Third Choice
3 Dep/3 Select/Blank23 Dep/3 Select
Probability (Tree Diagrams)
First Choice Second Choice
2 Dependent Events. 3 Selections
Third Choice
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 coloured 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 colour before replacing it. He then chooses a second cube at random. Record the information in a tree diagram.
Probability (Tree Diagrams)
Worksheet 1
Worksheet 2
Dependent
red
blue
First Choice Second Choice
red
blue
red
blue
The following situation can be represented by a tree diagram.
Peter has ten coloured cubes in a bag. Three of the cubes are red and seven are blue. He removes a cube at random from the bag and notes the colour but does not replace it. He then chooses a second cube at random. Record the information in a tree diagram.
Probability (Tree Diagrams)
Dependent Events