1. TREES AND COUNTING TECHNIQUES When we simple events like rolling a die, the number of possible outcomes is easily found. When you start to have multiple or complex events occurring, finding the total number of outcomes can become difficult. Trees and counting techniques help us find the total number of outcomes.

2. TREES AND COUNTING TECHNIQUES

5. A tree diagram gives a visual display of the total number of outcomes of an experiment consisting of a series of events. From a tree diagram, we can also see possible individual outcomes.

6. TREES AND COUNTING TECHNIQUES A tree diagram gives a visual display of the total number of outcomes of an experiment consisting of a series of events. From a tree diagram, we can also see possible individual outcomes. Using our last example : start Psychology sections 1 2 Sherry had 2 sections of psychology to choose from so we have 2 branches to start…

7. TREES AND COUNTING TECHNIQUES A tree diagram gives a visual display of the total number of outcomes of an experiment consisting of a series of events. From a tree diagram, we can also see possible individual outcomes. Using our last example : start Psychology sections 1 2 Sherry had 2 sections of physiology to choose from so we have 2 branches coming off EACH of the psychology paths… Physiology sections 1 1 2 2

8. TREES AND COUNTING TECHNIQUES A tree diagram gives a visual display of the total number of outcomes of an experiment consisting of a series of events. From a tree diagram, we can also see possible individual outcomes. Using our last example : start Psychology sections 1 2 Sherry had 3 sections of Spanish II to choose from so we have 3 branches coming off EACH of the physiology paths… Physiology sections 1 1 2 2 1 1 1 1 2 2 2 2 3 3 3 3 Spanish sections

9. TREES AND COUNTING TECHNIQUES A tree diagram gives a visual display of the total number of outcomes of an experiment consisting of a series of events. From a tree diagram, we can also see possible individual outcomes. Using our last example : start Psychology sections 1 2 Sherry had 3 sections of Spanish II to choose from so we have 3 branches coming off EACH of the physiology paths… Physiology sections 1 1 2 2 1 1 1 1 2 2 2 2 3 3 3 3 Spanish sections If you count the number of end branches, we get 12 possible outcomes…

10. TREES AND COUNTING TECHNIQUES A tree diagram gives a visual display of the total number of outcomes of an experiment consisting of a series of events. From a tree diagram, we can also see possible individual outcomes. Using our last example : start Psychology sections 1 2 We also can see possible schedules… Physiology sections 1 1 2 2 1 1 1 1 2 2 2 2 3 3 3 3 Spanish sections

11. TREES AND COUNTING TECHNIQUES A tree diagram gives a visual display of the total number of outcomes of an experiment consisting of a series of events. From a tree diagram, we can also see possible individual outcomes. Using our last example : start Psychology sections 1 2 We also can see possible schedules… Psyche 1, Physio 2, Spanish 2 Physiology sections 1 1 2 2 1 1 1 1 2 2 2 2 3 3 3 3 Spanish sections

12. TREES AND COUNTING TECHNIQUES A tree diagram gives a visual display of the total number of outcomes of an experiment consisting of a series of events. From a tree diagram, we can also see possible individual outcomes. Using our last example : start Psychology sections 1 2 We also can see possible schedules… Psyche 1, Physio 2, Spanish 2 OR Psyche 2, Physio 1, Spanish 3 Physiology sections 1 1 2 2 1 1 1 1 2 2 2 2 3 3 3 3 Spanish sections

13. TREES AND COUNTING TECHNIQUES Tree diagrams and probability : Suppose there are 5 balls in an urn. They are identical except for color. Three of the balls are red and two of the balls are blue. You are instructed to draw one ball, note its color, and set it aside. Then you are to draw out another ball and note its color. a)Use a tree to find the possible outcomes of the experiment b)Find the probability of each outcome

14. TREES AND COUNTING TECHNIQUES Tree diagrams and probability : Suppose there are 5 balls in an urn. They are identical except for color. Three of the balls are red and two of the balls are blue. You are instructed to draw one ball, note its color, and set it aside. Then you are to draw out another ball and note its color. a)Use a tree to find the possible outcomes of the experiment b)Find the probability of each outcome Let’s first set up the tree diagram…

15. TREES AND COUNTING TECHNIQUES Tree diagrams and probability : Suppose there are 5 balls in an urn. They are identical except for color. Three of the balls are red and two of the balls are blue. You are instructed to draw one ball, note its color, and set it aside. Then you are to draw out another ball and note its color. a)Use a tree to find the possible outcomes of the experiment b)Find the probability of each outcome Let’s first set up the tree diagram… Blue Red Start Color of 1 st ball

16. TREES AND COUNTING TECHNIQUES Tree diagrams and probability : Suppose there are 5 balls in an urn. They are identical except for color. Three of the balls are red and two of the balls are blue. You are instructed to draw one ball, note its color, and set it aside. Then you are to draw out another ball and note its color. a)Use a tree to find the possible outcomes of the experiment b)Find the probability of each outcome Let’s first set up the tree diagram… Blue Red Start Blue Red Color of 1 st ball Color of 2 nd ball

17. TREES AND COUNTING TECHNIQUES Tree diagrams and probability : Suppose there are 5 balls in an urn. They are identical except for color. Three of the balls are red and two of the balls are blue. You are instructed to draw one ball, note its color, and set it aside. Then you are to draw out another ball and note its color. a)Use a tree to find the possible outcomes of the experiment b)Find the probability of each outcome Let’s first set up the tree diagram… Blue Red Start Blue Red We have 4 possible outcomes … Blue, Blue Blue, Red Red, Blue Red, Red Color of 1 st ball Color of 2 nd ball

18. TREES AND COUNTING TECHNIQUES Tree diagrams and probability : Suppose there are 5 balls in an urn. They are identical except for color. Three of the balls are red and two of the balls are blue. You are instructed to draw one ball, note its color, and set it aside. Then you are to draw out another ball and note its color. a)Use a tree to find the possible outcomes of the experiment b)Find the probability of each outcome Let’s first set up the tree diagram… Blue Red Start Blue Red Now let’s find the probabilities… Color of 1 st ball Color of 2 nd ball

19. TREES AND COUNTING TECHNIQUES Tree diagrams and probability : Suppose there are 5 balls in an urn. They are identical except for color. Three of the balls are red and two of the balls are blue. You are instructed to draw one ball, note its color, and set it aside. Then you are to draw out another ball and note its color. a)Use a tree to find the possible outcomes of the experiment b)Find the probability of each outcome Let’s first set up the tree diagram… Blue Red Start Blue Red Now let’s find the probabilities… The first paths are easy, we have 5 total objects, 2 are blue and 3 are red. Color of 1 st ball Color of 2 nd ball

20. TREES AND COUNTING TECHNIQUES Tree diagrams and probability : Suppose there are 5 balls in an urn. They are identical except for color. Three of the balls are red and two of the balls are blue. You are instructed to draw one ball, note its color, and set it aside. Then you are to draw out another ball and note its color. a)Use a tree to find the possible outcomes of the experiment b)Find the probability of each outcome Let’s first set up the tree diagram… Blue Red Start Blue Red Now let’s find the probabilities… The first paths are easy, we have 5 total objects, 2 are blue and 3 are red On the second paths we did not replace the ball, so now there are only 4 balls left; 1 blue and 3 red if blue was drawn on the 1 st draw, and 2 red and 2 blue if red was the 1 st draw. Color of 1 st ball Color of 2 nd ball

21. TREES AND COUNTING TECHNIQUES Tree diagrams and probability : Suppose there are 5 balls in an urn. They are identical except for color. Three of the balls are red and two of the balls are blue. You are instructed to draw one ball, note its color, and set it aside. Then you are to draw out another ball and note its color. a)Use a tree to find the possible outcomes of the experiment b)Find the probability of each outcome Let’s first set up the tree diagram… Blue Red Start Blue Red Color of 1 st ball Color of 2 nd ball

22. TREES AND COUNTING TECHNIQUES Tree diagrams and probability : Suppose there are 5 balls in an urn. They are identical except for color. Three of the balls are red and two of the balls are blue. You are instructed to draw one ball, note its color, and set it aside. Then you are to draw out another ball and note its color. a)Use a tree to find the possible outcomes of the experiment b)Find the probability of each outcome Let’s first set up the tree diagram… Blue Red Start Blue Red Color of 1 st ball Color of 2 nd ball

23. TREES AND COUNTING TECHNIQUES Tree diagrams and probability : Suppose there are 5 balls in an urn. They are identical except for color. Three of the balls are red and two of the balls are blue. You are instructed to draw one ball, note its color, and set it aside. Then you are to draw out another ball and note its color. a)Use a tree to find the possible outcomes of the experiment b)Find the probability of each outcome Let’s first set up the tree diagram… Blue Red Start Blue Red Color of 1 st ball Color of 2 nd ball

24. TREES AND COUNTING TECHNIQUES Tree diagrams and probability : Suppose there are 5 balls in an urn. They are identical except for color. Three of the balls are red and two of the balls are blue. You are instructed to draw one ball, note its color, and set it aside. Then you are to draw out another ball and note its color. a)Use a tree to find the possible outcomes of the experiment b)Find the probability of each outcome Let’s first set up the tree diagram… Blue Red Start Blue Red Color of 1 st ball Color of 2 nd ball

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29. Counting rule for Permutations What if you are having a party for eight people and you have a table that seats five. How many different ways could your guests seat themselves at that table ?

30. TREES AND COUNTING TECHNIQUES

33. Counting rule for Combinations With permutations we are considering groupings and order. With Combinations, we don’t care about the order, just the groupings. For example : John is taking AP Literature and has to read four novels from a list of ten over the summer. How many groups of four could we have from a list of ten books ?

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