Tree Diagram

Tree Diagram

A tree diagram is a visual representation of all possible future outcomes and the associated probabilities of a random variable. Tree diagrams are particularly useful when we have several possible outcomes. They can help you record all the possibilities in a clear, uncomplicated manner. Each branch in a tree diagram represents an outcome. Consider the following example:

Example

If we tossed a fair coin twice, the first outcome would not affect the second outcome (the two events are independent). The outcome of the first toss can either be a head or a tail. Similarly, the outcome of the second toss can be a head or a tail, regardless of the outcome of the first toss, since the coin has no memory! We can represent the probabilities in a tree diagram as shown below.

tree-diagram

Please note the following:

  • the tree diagram must include all the possible outcomes;
  • the sum of the probabilities must add up to 1;
  • the number of branches is the number of different possibilities; and
  • probabilities are represented by the numbers on the branches.

To calculate probabilities, we go along the tree branches from left to right to get to the end and then multiply any probabilities that we have passed together. Therefore, the probability of getting two heads, i.e., HH, is:

$$ \begin{align*}
\text P(\text{HH})& =\cfrac {1}{2} * \cfrac {1}{2} \\
&= \cfrac {1}{4} \\
\end{align*} $$

If we sum up the probabilities of all possibilities, we get 1. (The candidate can prove this.)

Question

A card is picked from an ordinary pack of 52 playing cards, without replacement, and then another one is picked. Draw a probability tree and use it to calculate the probability of picking two red cards.

A. 25/102

B. 13/51

C. 26/51

Solution

The correct answer is A.

tree-diagram-example

Reading 8 LOS 8j

Explain the use of a tree diagram to represent an investment problem.

Shop CFA® Exam Prep

Offered by AnalystPrep

Featured Shop FRM® Exam Prep Learn with Us

    Subscribe to our newsletter and keep up with the latest and greatest tips for success
    Shop Actuarial Exams Prep Shop Graduate Admission Exam Prep


    Sergio Torrico
    Sergio Torrico
    2021-07-23
    Excelente para el FRM 2 Escribo esta revisión en español para los hispanohablantes, soy de Bolivia, y utilicé AnalystPrep para dudas y consultas sobre mi preparación para el FRM nivel 2 (lo tomé una sola vez y aprobé muy bien), siempre tuve un soporte claro, directo y rápido, el material sale rápido cuando hay cambios en el temario de GARP, y los ejercicios y exámenes son muy útiles para practicar.
    diana
    diana
    2021-07-17
    So helpful. I have been using the videos to prepare for the CFA Level II exam. The videos signpost the reading contents, explain the concepts and provide additional context for specific concepts. The fun light-hearted analogies are also a welcome break to some very dry content. I usually watch the videos before going into more in-depth reading and they are a good way to avoid being overwhelmed by the sheer volume of content when you look at the readings.
    Kriti Dhawan
    Kriti Dhawan
    2021-07-16
    A great curriculum provider. James sir explains the concept so well that rather than memorising it, you tend to intuitively understand and absorb them. Thank you ! Grateful I saw this at the right time for my CFA prep.
    nikhil kumar
    nikhil kumar
    2021-06-28
    Very well explained and gives a great insight about topics in a very short time. Glad to have found Professor Forjan's lectures.
    Marwan
    Marwan
    2021-06-22
    Great support throughout the course by the team, did not feel neglected
    Benjamin anonymous
    Benjamin anonymous
    2021-05-10
    I loved using AnalystPrep for FRM. QBank is huge, videos are great. Would recommend to a friend
    Daniel Glyn
    Daniel Glyn
    2021-03-24
    I have finished my FRM1 thanks to AnalystPrep. And now using AnalystPrep for my FRM2 preparation. Professor Forjan is brilliant. He gives such good explanations and analogies. And more than anything makes learning fun. A big thank you to Analystprep and Professor Forjan. 5 stars all the way!
    michael walshe
    michael walshe
    2021-03-18
    Professor James' videos are excellent for understanding the underlying theories behind financial engineering / financial analysis. The AnalystPrep videos were better than any of the others that I searched through on YouTube for providing a clear explanation of some concepts, such as Portfolio theory, CAPM, and Arbitrage Pricing theory. Watching these cleared up many of the unclarities I had in my head. Highly recommended.