Tree diagrams (Träddiagram)

Ämne / Topic

Drawing tree diagrams and using them to calculate probability of multiple events.

Inlärningsmål / Learning Goals

Understand how to draw tree diagrams and use them to calculate the probability for multiple events.

Google Slides & Videos

3. Aktiviteter / Key Activities

What is a Probability Tree Diagram?

We can construct a probability tree diagram to help us solve some probability problems.

A probability tree diagram shows all the possible events. The first event is represented by a dot. From the dot, branches are drawn to represent all possible outcomes of the event. The probability of each outcome is written on its branch.

Example:

A bag contains 3 black balls and 5 white balls. Paul picks a ball at random from the bag and replaces it back in the bag. He mixes the balls in the bag and then picks another ball at random from the bag.

a) Construct a probability tree of the problem.

b) Calculate the probability that Paul picks:

i) two black balls

ii) a black ball in his second draw

Solution:

Example:

Bag A contains 10 marbles of which 2 are red and 8 are black. Bag B contains 12 marbles of which 4 are red and 8 are black. A ball is drawn at random from each bag.

a) Draw a probability tree diagram to show all the outcomes the experiment.

b) Find the probability that:

(i) both are red.

(ii) both are black.

(iii) one black and one red.

(iv) at least one red.

Solution:

a) A probability tree diagram that shows all the outcomes of the experiment.

Example:

A box contains 4 red and 2 blue chips. A chip is drawn at random and then replaced. A second chip is then drawn at random.

a) Show all the possible outcomes using a probability tree diagram.

b) Calculate the probability of getting:

(i) at least one blue.

(ii) one red and one blue.

(iii) two of the same color.

Solution:

a) A probability tree diagram to show all the possible outcomes.

Probability Tree Diagrams for Independent Events

How to solve probability problems using probability tree diagrams?

Example:

A coin is biased so that it has a 60% chance of landing on heads. If it is thrown three times, find the probability of getting

a) three heads

b) 2 heads and a tail

c) at least one head

• Show Step-by-step Solutions

Probability Tree Diagrams for Dependent Events

How to use a probability tree diagram to calculate probabilities of two events which are not independent?

Example: Jimmy has a bag with seven blue sweets and 3 red sweets in it. He picks up a sweet at random from the bag, but does not replaces it and then picks again at random. Draw a tree diagram to represent this situation and use it to calculate the probabilities that he picks:

(a) two red sweets

(b) no red sweets

(c) at least one blue sweet

(d) one sweet of each color

• Show Step-by-step Solutions

4. Kolla vad eleverna kan / Check for understanding / Exit ticket

1. You roll a die twice and add up the dots to get a score. Draw a tree diagram to represent this experiment. What is the probability that your score is a multiple of 5?

2. What is the probability of throwing at least one five in four rolls of a regular

6-sided die? Hint: do not show all possible outcomes of each roll of the die. We are interested in whether the outcome is

5or not 5only

3. You flip one coin 4 times

times.

a. What is the probability of getting exactly 3 heads?

b. What is the probability of getting at least 3 heads?

5. Stängning / Closure / Summary of lesson

Review why we use tree diagrams.

Resurser & Materiel / Resources & Materials

http://webbapp.liber.se/matematikboken-z/#

http://webbapp.liber.se/matematikboken-z/#/5-sannolikhet-statistik

Textbook Practice

Ett pg. 228 no. 5022 abcd, 5023 abcd.

Två pg.229 no. 5026 abcd, 5027 abcd.

Tre pg. 230 no. 5030 ab, 5031 ab.

Fyra pg. 231 no. 5033 ab, 5034, 5035.