tree diagrams

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

HT

T

T

HH

HT

TH

TT

2nd 1st

Possible Outcomes

Attach probabilities

H

H

HT

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

HT

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=¼Probability 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

RR2nd 1st

B

B

BR

R

R

RG

G

G

GRB

RGBR

BB

BGGRGB

GG

INDEPENDENT EVENTS

B

RR2nd 1st

B

B

BR

R

R

RG

G

G

GRB

RGBR

BB

BGGRGB

GG

0.3

0.2

0.5

0.5

0.20.30.5

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.


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



1st Pick 2nd Pick

R

R

R

B

B

B

Probability Trees Example 1A 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



1st Pick 2nd Pick

R

R

R

B

B

B

610

410

59

49

69

39

?

?

??

?

?

To Part (ii)





1st Pick 2nd Pick

R

R

R

B

B

B

610

410

59

49

69

39

6 5 30( , )10 9 90

P R R





1st Pick 2nd Pick

R

R

R

B

B

B

610

410

59

49

69

39

6 4 24( , )10 9 90

P R B

4 6 24( , )10 9 90

P B R

24 24 48( )90 90 90

P oneofeach

Probability Trees Question 1

A bag contains 7 red beads and 3 blues. 2 beads are picked at random without replacement.








1st Pick 2nd Pick

R

R

R

B

B

B

710

310

69

39

79

29 To Part (ii)

?

?

??

?

?





1st Pick 2nd Pick

R

R

R

B

B

B

710

310

69

39

79

29 To Part (iii)

7 6 42( , )10 9 90

P R R





1st Pick 2nd Pick

R

R

R

B

B

B

710

310

69

39

79

29

7 3 21( , )10 9 90

P R B

3 7 21( , )10 9 90

P B R

21 21 42(one of each)90 90 90

P


A bag contains 4 yellow beads and 3 blues. 2 beads are picked at random without replacement.


(ii) Calculate the probability that both beads selected will be blue


Probability Trees Solution 2

A bag contains 4 yellow beads and 3 blues. 2 beads are picked at random without replacement.


(ii) Calculate the probability that both beads selected will be blue


1st Game 2nd Game

BB

B

Y

Y

Y47

3 2 6( , )7 6 42

P B B

4 3 12( , )7 6 42

P Y B 46

463

726

36

3 4 12( , )7 6 42

P B Y

12 12 24(One of each)42 42 42

P


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

RR

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

RR

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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