Odds

Odds offer another method to quantify confidence about the outcomes of an event, frequently employed in areas like sports where the comparison between an event and its complement is crucial. For instance, analyzing whether a team will win or lose a game or if a particular runner will finish a race first.

The odds in favor of an event happening are calculated by the ratio of the probability of the event occurring to the probability that it will not occur. This can be represented algebraically as such:

Where:

• \(P(A)\) is the probability of event \(A\)
• \(P(A')\) is the probability of the complement of event \(A\)

This expression is a common method to quantify probabilities.

Consider a card game where you are drawing from a shuffled deck of 52 cards, and you want to calculate the odds of drawing an Ace.

Let event \(A\) be drawing an Ace. Since there are 4 Aces in the deck, we can use the Probability formula to determine the likelihood of drawing this card:

\(P(A) = \cfrac{n(A)}{n(S)}\)

\(P(A) = \cfrac{4}{52}\)

\(P(A) = \cfrac{1}{13}\)

Next, we can determine the probability of not drawing an Ace, \(P(A')\), as such:

\(P(A') = 1 - P(A)\)

\(P(A') = 1 - \cfrac{1}{13}\)

\(P(A') = \cfrac{12}{13}\)

Then, we can use the Odds formula to determine the odds of pulling an Ace:

\(\text{Odds in favor of } \; A = \cfrac{P(A)}{P(A')}\)

\(\text{Odds in favor of } \; A = \cfrac{\cfrac{1}{13}}{\cfrac{12}{13}}\)

\(\text{Odds in favor of } \; A = \cfrac{1}{\cancel{13}} \times \cfrac{\cancel{13}}{12}\)

\(\text{Odds in favor of } \; A = \cfrac{1}{12}\)

Therefore, the odds in favor of drawing an Ace are 1:12, meaning it is much more likely that an Ace will not be drawn.

A variation often used is expressing the odds against an event. For the above example, the odds against drawing an Ace would be:

This means the odds against drawing an Ace are 12:1.