MATHEMATICSProbability – Tree Diagrams, OR and AND

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

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.

Scenario 15 red beads, 3 black beads - replacement

1st bead Final 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 = 45/512BBR = 3/8 × 3/8 × 5/8 = 45/512

BBB = 3/8 × 3/8 × 3/8 = 27/512

5/8

Black

Red

3/8

2nd bead 3rd 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

Scenario 25 red beads, 3 black beads – NO replacement

1st bead Final 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 = 30/336BBR = 3/8 × 2/7 × 5/6 = 30/336

BBB = 3/8 × 2/7 × 1/6 = 6/336

5/8

Black

Red

3/8

2nd bead 3rd 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

3/6

4/6

2/6

4/6

2/6

5/6

1/6

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

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

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.

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.