1. MATHEMATICS Probability – Tree Diagrams, OR and AND

2. Lesson Objectives The aim of this powerpoint is to help you… complete and interpret tree diagrams practice using the OR and AND rules for probability

3. Scenarios 1 & 2 For both of the following scenarios, we are going to have a bag containing 5 red beads and 3 black beads. We are going to take 3 beads out of the bag BUT… The first scenario is going to involve re-placing the bead we take out back into the bag each time, so none of the probabilities will change. The second scenario is going to involve not replacing the bead(s) in the bag, thus leaving fewer beads in the bag for the next selection…so all the probabilities WILL change.

4. Scenario 1 5 red beads, 3 black beads - replacement 1 st beadFinal Outcomes: RRR = 5/8 × 5/8 × 5/8 = 125/512 RRB = 5/8 × 5/8 × 3/8 = 75/512 RBR = 5/8 × 3/8 × 5/8 = 75/512 RBB = 5/8 × 3/8 × 3/8 = 45/512 BRR = 3/8 × 5/8 × 5/8 = 75/512 BRB = 3/8 × 5/8 × 3/8 = 45/512 BBR = 3/8 × 3/8 × 5/8 = 45/512 BBB = 3/8 × 3/8 × 3/8 = 27/512 5/8 Black Red 3/8 2 nd bead3 rd bead Black Red Black Red Black Red Black Red Black Red Black Red CHECK: All the probabilities in red add up to 512/512 = 1 5/8 3/8 5/8 3/8 5/8 3/8 5/8 3/8 5/8 3/8 5/8 3/8

5. Scenario 2 5 red beads, 3 black beads – NO replacement 1 st beadFinal Outcomes: RRR = 5/8 × 4/7 × 3/6 = 60/336 RRB = 5/8 × 4/7 × 3/6 = 60/336 RBR = 5/8 × 3/7 × 4/6 = 60/336 RBB = 5/8 × 3/7 × 2/6 = 30/336 BRR = 3/8 × 5/7 × 4/6 = 60/336 BRB = 3/8 × 5/7 × 2/6 = 30/336 BBR = 3/8 × 2/7 × 5/6 = 30/336 BBB = 3/8 × 2/7 × 1/6 = 6/336 5/8 Black Red 3/8 2 nd bead3 rd bead Black Red Black Red Black Red Black Red Black Red Black Red CHECK: All the probabilities in red add up to 336/336 = 1 4/7 3/7 5/7 2/7 3/6 4/6 2/6 4/6 2/6 5/6 1/6

6. Question 1 What is the probability of pulling out 2 reds and a black bead (in any order) for each scenario? Scenario 1: P(RRB) or P(RBR) or P(BRR) 75/512 + 75/512 + 75/512 = 225/512 Scenario 1: P(RRB) or P(RBR) or P(BRR) 60/336 + 60/336 + 60/336 = 180/336

7. Question 2 What is the probability of pulling out 3 beads of the same colour for each scenario? Scenario 1: P(RRR) or P(BBB) 125/512 + 27/512 = 152/512 Scenario 1: P(RRR) or P(BBB) 60/336 + 6/336 = 66/336

8. Individual Practice & Notes Please work through the worksheet called: ProbTree-S2.xlsx When you have finished this worksheet, ask your teacher to mark your work. In your exercise books or on paper, please write up your own notes and examples on tree diagrams. Ensure you include details of how NON-REPLACEMENT affects the probabilities on subsequent branches.

9. What next? If you have completed everything so far and there is still some time left, then it would be very useful to undertake the following two MyMaths lessons… MyMaths > Data > Probability > Venn Diagrams 1 MyMaths > Data > Probability > Venn Diagrams 2 BOTH of these MyMaths lessons are useful to ALL students whether or not they did the extension work on tree diagrams and the OR and AND rules.