Probability Tree Diagrams



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.

In this lesson, we will learn

  • how to draw probability tree diagrams for independent events (with replacement)
  • how to draw probability tree diagrams for dependent events (without replacement)

Related Topics:
More Probability Lessons

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:

a)

Check that the probabilities in the last column add up to 1.

b) i) To find the probability of getting two black balls, first locate the B branch and then follow the second B branch. Since these are independent events we can multiply the probability of each branch.


ii) There are two outcomes where the second ball can be black.

Either (B, B) or (W, B)


From the probability tree diagram, we get:

P(second ball black)

= P(B, B) or P(W, B)

= P(B, B) + P(W, B)

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.


b) The probability that:
(i) both are red.

P(R, R) =

(ii) both are black.

P(B, B) =

(iii) one black and one red.

P(R, B) or P(B, R) =

(iv) at least one red.

1- P(B, B) =

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.


b) The probability of getting:

(i) at least one blue.

P(R, B) or P(B, R) or P(B, B) =

(ii) one red and one blue.

P(R, B) or P(B, R) =

(iii) two of the same colour.

P(R, R) or P(B, B) =



The following videos give more examples of solving probability problems using probability tree diagrams.

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



How to use a tree diagram to calculate combined probabilities of two independent events



Using a probability tree diagram to calculate probabilities of two events which are not independent.

How to do draw accurate probability tree diagrams.

A bag contains 5 red sweets and 3 blue sweets. Two sweets are drawn at random
i) with replacement and II) without replacement.
Draw a tree diagram to represent the probabilities in each case.





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