Tree Diagrams



In this lesson, we will look at more examples of probabilty word problems
We will use tree diagrams to help us solve the problems. We will see that tree diagrams can be used to represent the set of all possible outcomes involving one or more experiments. Probability tree diagrams are useful for both independent (or unconditional) probability and dependent (or conditional) probability.

Related Topics:

More Tree Diagrams

More Probability Topics

Tree Diagrams for Independent Events

Example:

Julia spins 2 spinners; one of which is labelled 1, 2 and 3, and the other is labelled 4, 5 and 6.



a) Draw a tree diagram for the experiment.
b) What is the probability that the spinners stop at “3” and “4”?
c) Find the probability that the spinners do not stop at “3” and “4”.
d) What is the probability that the first spinner does not stop at “1”?

Solution:

a) Tree diagram for the experiment.


b) The probability that the spinners stop at “3” and “4”

n(S ) = 9
Probability that the spinners stop at (3,4) =

c) The probability that the spinners do not stop at “3” and “4”

Probability that the spinners do not stop at (3,4) =

d) The probability that the first spinner does not stop at “1”

Probability that the first spinner stop at “1” =
Probability that the first spinner does not stop at “1” =



Example:

Box A contains 3 cards numbered 1, 2 and 3. Box B contains 2 cards numbered 1 and 2. One card is removed at random from each box.

a) Draw a tree diagram to list all the possible outcomes.
b) Find the probability that:
(i) the sum of the numbers is 4
(ii) the sum of the two numbers is even.
(iii) the product of the two numbers is at least 5.
(iv) the sum is equal to the product.

Solution:

a) A tree diagram of all possible outcomes.


b) The probability that:

(i) the sum of the numbers is 4.

Let S be the sample space and A be the event that the sum is 4.

n(S) = 6; n(A) = 2

P(A) =

(ii) the sum of the two numbers is even.

Let B be the event that the sum is even.

n(B) = 3

P(B) =

(iii) the product of the two numbers is at least 5.

Let C be the event that the product of the two numbers is at least 5.

n(C) = 1

P(C) =

(iv) the sum is equal to the product.

Let D be the event that the sum of the two numbers is equal to the product.

n(D) = 1

P(D) =


Example:

A bag contains 4 cards numbered 2, 4, 6, 9. A second bag contains 3 cards numbered 2, 3, 6. One card is drawn at random from each bag.

a) Draw a tree diagram for the experiment.
b) With the help of the tree diagram, calculate the probability that the two numbers obtained:
(i) have different values.
(ii) are both even.
(iii) are both prime.
(iv) have a sum greater than 5.
(v) have a product greater than 16.

Solution:

a) A tree diagram of all possible outcomes.


b) The probability that the two numbers obtained:

(i) have different values.

Let S be the sample space and A be the event that the two values are different

n(S) = 12 ; n(A) = 10

P(A) =

(ii) are both even.

Let B be the event that both values are even.

n(B) = 6

P(B) =

(iii) are both prime.

Let C be the event that both values are prime.

n(C) = 2

P(C) =

(iv) have a sum greater than 5.

Let D be the event that the sum of both values is greater than 5.

n(D) = 10

P(D) =

(v) have a product greater than 16.

Let E be the event that the product of both values is greater than 16.

n(E) = 6

P(E) =


Using probability tree diagrams for independent events (or unconditional probability).

Jenny has a bag with 7 blue sweets and 3 red sweets. She picks a sweet at random from the bag, replaces it and picks again at random. Draw a tree diagram to represent this situation and use it to calculate the probabilities that she picks
a) 2 red sweets
b) no red sweets
c) at least 1 blue sweet
d) 1 sweet of each color



Probability Trees: Independent Events
How to do a probability tree. Multiplying and adding probabilities of independent events
Problem: Sarah picks amrble from a bag. There are 8 marbles in the bag and 5 of them are green. Ther rest are red. She looka at the marble and then places it into the bag. She then picks another marble. Complete a probability tree.
.




The following videos gives more examples of solving probability problems using tree diagrams. A tree diagram can help you generate all the outcomes.






Tree Diagrams for Dependent Events

Using probability tree diagrams for dependent events (or conditional probability).

Jimmy has a bag with 7 blue sweets and 3 red sweets. He picks a sweet at random from the bag, but does NOT replace it and picks again at random. Draw a tree diagram to represent this situation and use it to calculate the probabilities that she picks
a) 2 red sweets
b) no red sweets
c) at least 1 blue sweet
d) 1 sweet of each color



Probability Diagrams for events that involve with and without replacements.

A bag contains 4 red sweets and 5 blue sweets. Draw a probability tree diagram when
a) the sweets are taken with replacement
b) the sweets are taken without replacement








Tree Diagrams for Dependent Events
Use a probability tree diagram to calculate probabilities of two events which are not independent.
Jenny has a bag with 7 blue sweets and 3 red sweets. She picks a sweet at random from the bag, but does not replace it and picks again at random. Draw a tree diagram to represent this situation and use it to calculate the probabilities that she picks
a) 2 red sweets
b) no red sweets
c) at least 1 blue sweet
d) 1 sweet of each color



Probability: A bag contains four light bulbs, of which two are defective. We draw bulbs without replacement until a working bulb is selected. Set up the tree diagram for this experiment, find the probability of each outcome, and determine the probability that at most two draws occur.







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