Tree diagrams
What are Tree Diagrams
A way of showing the possibilities of two or more events
Simple diagram we use to calculate the probabilities of two or more events
A fair coin is spun twice
H
H
H
T
T
T
HH
HT
TH
TT
2nd 1st
Possible Outcomes
Attach probabilities
H
H
H
T
T
T
HH
HT
TH
TT
2nd 1st
½
½
½
½
½
½
P(H,H)=½x½=¼
P(H,T)=½x½=¼
P(T,H)=½x½=¼
P(T,T)=½x½=¼
INDEPENDENT EVENTS – 1st spin has no effect on the 2nd spin
Calculate probabilities
H
H
H
T
T
T
HH
HT
TH
TT
2nd 1st
½
½
½
½
½
½
P(H,H)=½x½=¼
P(H,T)=½x½=¼
P(T,H)=½x½=¼
P(T,T)=½x½=¼
Probability of at least one Head?
*
**
For example – 10 coloured beads in a bag – 3 Red, 2 Blue, 5 Green. One taken, colour noted, returned to bag, then a second taken.
B
RR
2nd 1st
B
B
BR
R
R
R
G
G
G
G
RB
RGBR
BB
BGGR
GB
GG
INDEPENDENT EVENTS
B
RR
2nd 1st
B
B
BR
R
R
R
G
G
G
G
RB
RGBR
BB
BGGR
GB
GG
0.3
0.2
0.5
0.5
0.20.3
0.5
0.20.3
0.5
0.20.3
Probabilities
P(RR) = 0.3x0.3 = 0.09
P(RB) = 0.3x0.2 = 0.06
P(RG) = 0.3x0.5 = 0.15P(BR) = 0.2x0.3 = 0.06
P(BB) = 0.2x0.2 = 0.04
P(BG) = 0.2x0.5 = 0.10P(GR) = 0.5x0.3 = 0.15
P(GB) = 0.5x0.2 = 0.10
P(GG) = 0.5x0.5 = 0.25
All ADD UP to 1.0
Tree DiagramsCould make a list Could draw up a table
Probability of two or more events
1st Throw 2nd Throw
THHHHH TTTT 1/21/21/21/21/21/21/2
OUTCOMES
H,H
H,T
T,H
T,T
P(Outcome)
P(H,H)=1/2x1/2=1/4
P(H,T)=1/2x1/2=1/4
P(T,H)=1/2x1/2=1/4
P(T,T)=1/2x1/2=1/4
Total P(all outcomes) = 1
3/9
6/9
7/10
3/10
2/9
7/9
1st event2nd event
7 Red 3 Blue. Pick 2, without replacement. a) p(R,R) b) p(B,B) c) p(One of each)
OUTCOMES P(Outcome)
R,R
R,B
B,R
B,B
P(R,R)=42/90
P(R,B)=21/90
P(B,R)=21/90
P(B,B)=6/90
Total P(all outcomes) = 1
Probability Trees Example 1
A bag contains 6 red beads and 4 blues. 2 beads are picked at random without replacement.
(i) Draw a probability tree diagram to show this information
(ii) Calculate the probability of selecting both red beads
(iii) Calculate the probability of picking one of each colour.
Probability Trees Example 1
A bag contains 6 red beads and 4 blues. 2 beads are picked at random without replacement. (i) Draw a probability tree diagram to show this information
(ii) Calculate the probability of selecting both red beads
(iii) Calculate the probability of picking one of each colour.
1st Pick 2nd Pick
R
R
R
B
B
B
Probability Trees Example 1
A bag contains 6 red beads and 4 blues. 2 beads are picked at random without replacement. (i) Draw a probability tree diagram to show this information
(ii) Calculate the probability of selecting both red beads
(iii) Calculate the probability of picking one of each colour.
1st Pick 2nd Pick
R
R
R
B
B
B
6
10
4
10
5
9
4
9
6
9
3
9
?
?
??
?
?
To Part (ii)
Probability Trees Example 1
A bag contains 6 red beads and 4 blues. 2 beads are picked at random without replacement. (i) Draw a probability tree diagram to show this information
(ii) Calculate the probability of selecting both red beads
(iii) Calculate the probability of picking one of each colour.
1st Pick 2nd Pick
R
R
R
B
B
B
6
10
4
10
5
9
4
9
6
9
3
9
6 5 30( , )
10 9 90P R R
Probability Trees Example 1
A bag contains 6 red beads and 4 blues. 2 beads are picked at random without replacement. (i) Draw a probability tree diagram to show this information
(ii) Calculate the probability of selecting both red beads
(iii) Calculate the probability of picking one of each colour.
1st Pick 2nd Pick
R
R
R
B
B
B
6
10
4
10
5
9
4
9
6
9
3
9
6 4 24( , )
10 9 90P R B
4 6 24( , )
10 9 90P B R
24 24 48( )
90 90 90P oneofeach
Probability Trees Question 1
A bag contains 7 red beads and 3 blues. 2 beads are picked at random without replacement.
(i) Draw a probability tree diagram to show this information
(ii) Calculate the probability of selecting both red beads
(iii) Calculate the probability of picking one of each colour.
Probability Trees Question 1
A bag contains 7 red beads and 3 blues. 2 beads are picked at random without replacement. (i) Draw a probability tree diagram to show this information
(ii) Calculate the probability of selecting both red beads
(iii) Calculate the probability of picking one of each colour.
1st Pick 2nd Pick
R
R
R
B
B
B
7
10
3
10
6
9
3
9
7
9
2
9 To Part (ii)
?
?
??
?
?
Probability Trees Question 1
A bag contains 7 red beads and 3 blues. 2 beads are picked at random without replacement. (i) Draw a probability tree diagram to show this information
(ii) Calculate the probability of selecting both red beads
(iii) Calculate the probability of picking one of each colour.
1st Pick 2nd Pick
R
R
R
B
B
B
7
10
3
10
6
9
3
9
7
9
2
9 To Part (iii)
7 6 42( , )
10 9 90P R R
Probability Trees Question 1
A bag contains 7 red beads and 3 blues. 2 beads are picked at random without replacement. (i) Draw a probability tree diagram to show this information
(ii) Calculate the probability of selecting both red beads
(iii) Calculate the probability of picking one of each colour.
1st Pick 2nd Pick
R
R
R
B
B
B
7
10
3
10
6
9
3
9
7
9
2
9
7 3 21( , )
10 9 90P R B
3 7 21( , )
10 9 90P B R
21 21 42(one of each)
90 90 90P
Probability Trees Question 2
A bag contains 4 yellow beads and 3 blues. 2 beads are picked at random without replacement.
(i) Draw a probability tree diagram to show this information
(ii) Calculate the probability that both beads selected will be blue
(iii) Calculate the probability of picking one of each colour.
Probability Trees Solution 2
A bag contains 4 yellow beads and 3 blues. 2 beads are picked at random without replacement.
(i) Draw a probability tree diagram to show this information
(ii) Calculate the probability that both beads selected will be blue
(iii) Calculate the probability of picking one of each colour.
1st Game 2nd Game
B
B
B
Y
Y
Y4
7
3 2 6( , )
7 6 42P B B
4 3 12( , )
7 6 42P Y B
4
6
4
63
7
2
6
3
63 4 12
( , )7 6 42
P B Y
12 12 24(One of each)
42 42 42P
Probability Trees Question 3
The probability that Stuart wins a game of darts against Rose is 0.7. They play two games.
(i) Copy & complete the probability tree diagram shown below
(ii) Calculate the probability Rose winning both games
(iii) Calculate the probability of the final result being a draw.
1st Game 2nd Game
R
R
R
S
S
S0.7
Probability Trees Solutions 3
The probability that Stuart wins a game of darts against Rose is 0.7. They play two games.
(i) Copy & complete the probability tree diagram shown below
(ii) Calculate the probability Rose winning both games
(iii) Calculate the probability of the final result being a draw.
1st Game 2nd Game
R
R
R
S
S
S0.7
0.3
0.7
0.7
0.3
0.3
( , ) 0.3 0.3 0.09P R R
( , ) 0.7 0.3 0.21P S R ( , ) 0.3 0.7 0.21P R S
( ) 0.42P Draw
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