###### Arithmetic Mean Return Vs Geometric Me ...

Both arithmetic return and geometric return are methods commonly used to calculate the... **Read More**

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)}{ \left\{ 1 – P(E) \right\} } \\ \\

\text{Odds against E} & = \cfrac { \left\{ 1 – P(E) \right\} }{ P(E) } \\

\end{align*} $$

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**

First, we have to establish the probability of drawing a blue ball:

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

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

$$ \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.

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

A. 2:1

B. 1:2

C. 1:1

SolutionThe correct answer is

C.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.