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 H H H T T T HH HT TH TT 2 nd 1 st Possible Outcomes

4. Attach probabilities H H H T T T HH HT TH TT 2 nd 1 st 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 – 1 st spin has no effect on the 2 nd spin

5. Calculate probabilities H H H T T T HH HT TH TT 2 nd 1 st 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 Probability of at least one Head? * * *

6. 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 RR 2 nd 1 st B B B R R R R G G G G RBRB RGRG BRBR BB BGBG GRGR GBGB GG INDEPENDENT EVENTS

7. B RR 2 nd 1 st B B B R R R R G G G G RBRB RGRG BRBR BB BGBG GRGR GBGB GG 0.3 0.2 0.5 0.2 0.3 0.5 0.2 0.3 0.5 0.2 0.3 Probabilities P(RR) = 0.3x0.3 = 0.09 P(RB) = 0.3x0.2 = 0.06 P(RG) = 0.3x0.5 = 0.15 P(BR) = 0.2x0.3 = 0.06 P(BB) = 0.2x0.2 = 0.04 P(BG) = 0.2x0.5 = 0.10 P(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

8. Tree Diagrams Could make a list Could draw up a table Probability of two or more events 1 st Throw 2 nd Throw T HHHH H T TTT 1/2 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

9. 3/9 6/9 7/10 3/10 2/9 7/9 1 st event 2 nd event 7 Red 3 Blue. Pick 2, without replacement. a) p(R,R) b) p(B,B) c) p(One of each) OUTCOMESP(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

10. 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.

11. 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. 1 st Pick2 nd Pick R R R B B B

12. 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. 1 st Pick2 nd Pick R R R B B B ? ? ? ? ? ? To Part (ii)

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. 1 st Pick2 nd Pick R R R B B B

14. 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. 1 st Pick2 nd Pick R R R B B B

15. 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.

16. 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. 1 st Pick2 nd Pick R R R B B B To Part (ii) ? ? ? ? ? ?

17. 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. 1 st Pick2 nd Pick R R R B B B To Part (iii)

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. 1 st Pick2 nd Pick R R R B B B

19. 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.

20. 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. 1 st Game 2 nd Game B B B Y Y Y

21. 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. 1 st Game 2 nd Game R R R S S S

22. 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. 1 st Game 2 nd Game R R R S S S