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 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.