1. Tree diagrams

2. 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

3. A fair coin is spun twice
1st
2nd H HH
Possible Outcomes H T HT H TH T T TT

4. Attach probabilities 1st 2nd H HH P(H,H)=½x½=¼ ½ ½ H ½ T HT
P(H,T)=½x½=¼ H TH
P(T,H)=½x½=¼ T T TT
P(T,T)=½x½=¼
INDEPENDENT EVENTS – 1st spin has no effect on the 2nd spin

5. * * * Calculate probabilities 1st 2nd H HH P(H,H)=½x½=¼ ½ ½ H ½ T HT
P(H,T)=½x½=¼ * H TH
P(T,H)=½x½=¼ T T TT
P(T,T)=½x½=¼
Probability of at least one Head?

6. 1st 2nd R RR B RB R G RG R BR B B BB G BG R GR G GB B G GG
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 RR B RB R G RG R BR
INDEPENDENT EVENTS B B BB G BG R GR G GB B G GG

7. Probabilities 1st 2nd R RR B RB R G RG R BR B B BB G BG R GR G GB B G
P(RR) = 0.3x0.3 = 0.09
0.3
0.2 B RB
P(RB) = 0.3x0.2 = 0.06 R
0.3
0.5 G RG
P(RG) = 0.3x0.5 = 0.15 R BR
P(BR) = 0.2x0.3 = 0.06
0.3
0.2
0.2 B B BB
P(BB) = 0.2x0.2 = 0.04
0.5 G BG
P(BG) = 0.2x0.5 = 0.10 R GR
0.3
P(GR) = 0.5x0.3 = 0.15
0.5 G
0.2 GB B
P(GB) = 0.5x0.2 = 0.10 G GG
0.5
P(GG) = 0.5x0.5 = 0.25
All ADD UP to 1.0

8. 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.
Slide 8

9. 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) =
P(black second) =
(There are 13 pens left and 5 are black)
THEREFORE………………………………………………
P(black, black) =
Slide 9

10. Dependent Events Find the probability 1 26 25 650 P(Q, Q)
All the letters of the alphabet are in the bag 1 time
Do not replace the letter 1 26 25
650 x =
Slide 10

11. Are these dependent or independent events?
TEST YOURSELF
Are these dependent or independent events?
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

12. 7 Red 3 Blue. Pick 2, without replacement
7 Red 3 Blue. Pick 2, without replacement. a) p(R,R) b) p(B,B) c) p(One of each)
OUTCOMES
P(Outcome)
2nd event
1st event
6/9
R,R
P(R,R)=42/90
7/10
3/10
3/9
R,B
P(R,B)=21/90
7/9
B,R
P(B,R)=21/90
2/9
B,B
P(B,B)=6/90
Total P(all outcomes) = 1

13. 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.

14. R R B R B B Probability Trees Example 1 1st Pick 2nd Pick
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

15. R R B R B B Probability Trees Example 1 1st Pick 2nd Pick ? ? ? ? ? ?
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
To Part (ii)

16. R R B R B B Probability Trees Example 1 1st Pick 2nd Pick
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

17. R R B R B B Probability Trees Example 1 1st Pick 2nd Pick
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

18. 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.

19. R R B R B B Probability Trees Question 1 1st Pick 2nd Pick ? ? ? ? ? ?
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 ? B ? ? R ? B ? B
To Part (ii)

20. R R B R B B Probability Trees Question 1 1st Pick 2nd Pick
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 B R B B
To Part (iii)

21. R R B R B B Probability Trees Question 1 1st Pick 2nd Pick
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 B R B B

22. 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.

23. Y Y B Y B B Probability Trees Solution 2 1st Game 2nd Game
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 Y Y B Y B B

24. Probability (Tree Diagrams)
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.
Probability (Tree Diagrams)
Dependent Events
Milk
Dark
First Pick
Second Pick
Q5 Chocolates

25. S S R S R R Probability Trees Question 3
The probability that Stuart wins a game of darts against Rose is 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 S R S R R

26. S S R S R R Probability Trees Solutions 3
The probability that Stuart wins a game of darts against Rose is 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 S R S R R

27. Independent Practice
Solve #1, 3, 4 on pages (Exercise 8P) For review (IB Test and non IB registered students) – Use exam style questions on pages 372 – 376.