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.

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 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

(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

610

410

59

49

69

39

?

?

??

?

?

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

610

410

59

49

69

39

6 5 30( , )10 9 90

P 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

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.

(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

710

310

69

39

79

29 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

710

310

69

39

79

29 To Part (iii)

7 6 42( , )10 9 90

P 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

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

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

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

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

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