Transcript Probability & Tree Diagrams

Probability & 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
For example – a fair coin is spun twice
1st
2nd
H
HH
T
HT
H
TH
T
TT
H
T
Possible
Outcomes
Attach probabilities
1st
½
½
2nd
½
H
H
HH
P(H,H)=½x½=¼
T
HT
P(H,T)=½x½=¼
½
H
TH
P(T,H)=½x½=¼
½
T
TT
P(T,T)=½x½=¼
½
T
INDEPENDENT EVENTS – 1st spin has no effect on the 2nd spin
Calculate probabilities
1st
½
½
2nd
½
H
H
HH
P(H,H)=½x½=¼
*
T
HT
P(H,T)=½x½=¼
½
H
TH
P(T,H)=½x½=¼
*
*
½
T
TT
P(T,T)=½x½=¼
½
T
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.
1st
2nd
R
B
G
R
RR
B
RB
G
R
RG
BR
B
BB
G
R
BG
GR
B
GB
G
GG
INDEPENDENT EVENTS
Probabilities
1st
2nd
0.3
0.2
0.3
R
0.5
0.3
0.2
0.2
B
0.5
0.5
0.3
0.2
G
0.5
R
RR
P(RR) = 0.3x0.3 = 0.09
B
RB
P(RB) = 0.3x0.2 = 0.06
G
R
RG
BR
P(RG) = 0.3x0.5 = 0.15
P(BR) = 0.2x0.3 = 0.06
B
BB
P(BB) = 0.2x0.2 = 0.04
G
R
BG
GR
P(BG) = 0.2x0.5 = 0.10
P(GR) = 0.5x0.3 = 0.15
B
GB
P(GB) = 0.5x0.2 = 0.10
G
GG
P(GG) = 0.5x0.5 = 0.25
All ADD UP to 1.0
For example – 10 coloured beads in a bag – 3 Red, 2 Blue,
5 Green. One taken, colour noted, NOT returned to bag,
then a second taken.
1st
2nd
R
B
G
R
RR
B
RB
G
R
RG
BR
B
BB
G
R
BG
GR
B
GB
G
GG
NOT INDEPENDENT EVENTS