Transcript MS Tree Diagrams without replacement

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
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
Dependent Event

What happens the during the second event
depends upon what happened before.

In other words, the result of the second event
will change because of what happened first.
The probability of two dependent events, A and B, is equal to the
probability of event A times the probability of event B. However,
the probability of event B now depends on event A.
P(A, B) = P(A)  P(B)
Slide 8
Dependent Event
Example: There are 6 black pens and 8 blue pens in a jar. If you
take a pen without looking and then take another pen without
replacing the first, what is the probability that you will get 2
black pens?
P(black first) =
6
3
or
14
7
5
P(black second) =
(There are 13 pens left and 5 are black)
13
THEREFORE………………………………………………
P(black, black) =
3 5
15

or
7 13
91
Slide 9
Dependent Events
Find the probability



P(Q, Q)
All the letters of the
alphabet are in the
bag 1 time
Do not replace the
letter
1
26
0
x
25
0
=
650
0
Slide 10
TEST YOURSELF
Are these dependent or independent events?
1.
Tossing two dice and getting a 6 on both of them.
2.
You have a bag of marbles: 3 blue, 5 white, and 12
red. You choose one marble out of the bag, look at it
then put it back. Then you choose another marble.
3.
You have a basket of socks. You need to find the
probability of pulling out a black sock and its matching
black sock without putting the first sock back.
4.
You pick the letter Q from a bag containing all the
letters of the alphabet. You do not put the Q back in the
bag before you pick another tile.
Slide 11
7 Red 3 Blue. Pick 2, without replacement. a) p(R,R) b) p(B,B) c) p(One of each)
OUTCOMES
2nd event
1st event
P(Outcome)
6/9
R,R
3/9
R,B
P(R,B)=21/90
7/9
B,R
P(B,R)=21/90
P(R,R)=42/90
7/10
3/10
2/9
B,B
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.
Example 1
Probability Trees
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
B
R
B
B
Example 1
Probability Trees
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
6
10
4
10
2nd Pick
5
9
R
4
9
B
6
9
R
3
9
B
R
B
To Part (ii)
Example 1
Probability Trees
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
6
10
4
10
2nd Pick
5
9
R
4
9
B
6
9
R
3
9
B
R
B
6 5 30
P ( R, R )   
10 9 90
Example 1
Probability Trees
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
5
9
6
10
4
10
R
R
4
9
B
6
9
3
9
B
6 4 24
P ( R, B )   
10 9 90
4 6 24
P ( B, R )   
10 9 90
24 24 48
P(oneofeach) 


90 90 90
R
B
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.
Question 1
Probability Trees
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
7
10
3
10
2nd Pick
6
9
R
3
9
B
7
9
R
2
9
B
R
B
To Part (ii)
Question 1
Probability Trees
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
7
10
3
10
2nd Pick
6
9
R
3
9
B
7
9
R
2
9
B
7 6 42
P ( R, R )   
10 9 90
R
B
To Part (iii)
Question 1
Probability Trees
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
6
9
7
10
3
10
R
R
3
9
B
7
9
2
9
B
7 3 21
P ( R, B )   
10 9 90
3 7 21
P ( B, R )   
10 9 90
21 21 42
P(one of each) 


90 90 90
R
B
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.
Solution 2
Probability Trees
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
4
7
3
7
2nd Game
Game
Y
B
3
6
4
6
Y
4
6
B
Y
2
6
B
3 2 6
P ( B, B )   
7 6 42
4 3 12
P(Y , B)   
7 6 42
3 4 12
P ( B, Y )   
7 6 42
12 12 24
P(One of each) 


42 42 42
Probability (Tree Diagrams)
Dependent Events
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.
Second Pick
First Pick
18
30
12
30
Milk
Dark
18 17 306
x

30 29 870
17
29
Milk P(milk and milk) =
12
29
Dark P(milk and dark) = 18 x 12  216
30
18
29
Milk P(dark and milk) =
11
29
Dark P(dark and dark) =
Q5 Chocolates
29
870
12 18 216
x

30 29 870
12 11 132
x

30 29 870
Question 3
Probability Trees
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
S
0.7
S
R
R
S
R
Solutions 3
Probability Trees
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
0.7
0.7
0.3
S
R
0.3
0.7
0.3
S
R
S
R
P( R, R)  0.3  0.3  0.09
P( S , R)  0.7  0.3  0.21
P( R, S )  0.3  0.7  0.21
P( Draw)  0.42
Independent Practice
Solve #1, 3, 4 on pages 370 -371 (Exercise 8P)
For review (IB Test and non IB registered
students) –
Use exam style questions on pages 372 – 376.