# Probability in Terms of Odds

Odds for and against an event represent a ratio of the desired outcomes versus the field. In other words, the odds for an event are the ratio of the number of ways the event can occur to the number of ways the event cannot occur.

Given the probability of an event ‘E’, i.e., P(E),

\begin{align*} \text{Odds for E} & =\cfrac {P(E)}{ 1 – P(E) } \\ \\ \text{Odds against E} & = \cfrac { 1 – P(E) }{ P(E) } \\ \end{align*}

#### Example: Odds for and Against an Event

A box contains five blue balls, two green balls, and six yellow balls. What are the odds of drawing a blue ball from the box?

Solution

Step 1: Establish the probability of drawing a blue ball:

Let P(B) represent the event that a blue ball is drawn from the box. Therefore,

$$P(B) = \cfrac {5}{13}$$

Step 2: Determine the odds for the blue ball as follows:

\begin{align*} \text{The odds for a blue ball} & =\cfrac {5}{13} ÷ \left(1 – \cfrac {5}{13} \right)\\ & = \cfrac {5}{13} ÷ \left( \cfrac {8}{13} \right) \\ & =\cfrac {5}{13} × \cfrac {13}{8} \\ & =\cfrac {5}{8} \\ \end{align*}

Therefore, the odds for a blue ball are 5:8 (pronounced as ‘5 to 8’).

Similarly, we can calculate the odds against drawing a blue ball:

\begin{align*} \text{Odds against a blue ball} &= \left\{1 – \cfrac{5}{13} \right\} ÷ \cfrac{5}{13} \\ &= \cfrac{8}{13} ÷ \cfrac{5}{13} \\ &= \cfrac{8}{13} × \cfrac{13}{5} \\ & = \cfrac{8}{5} \end{align*}

Therefore, the odds against drawing a blue ball are 8:5 (pronounced as ‘8 to 5’).

Note: You should notice that the odds against an event are always equal to the reciprocal of odds for the same event.

## Question

Suppose you toss a fair coin. What are the odds against obtaining a head?

A. 2:1.

B. 1:2.

C. 1:1.

Solution

The probability of obtaining a head, P(H) = 1/2. Therefore,

\begin{align*} \text{Odds against a head} & = \left\{ 1 – P(H) \right\} ÷ P(H) \\ & = \left\{ 1 – \cfrac{1}{2} \right\} ÷ \cfrac{1}{2} \\ & = \cfrac{1}{2} × \cfrac{2}{1} \\ & = \cfrac{1}{1} \end{align*}

Therefore, the odds against a head are 1:1, pronounced as ‘1 to 1.’

Note that in this case, the odds for a head are also 1:1.

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

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
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
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
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
2021-06-22
Great support throughout the course by the team, did not feel neglected
Benjamin anonymous
2021-05-10
I loved using AnalystPrep for FRM. QBank is huge, videos are great. Would recommend to a friend
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
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.