Odds Calculator

When making predictions, people often use the terms "probability" and "odds" interchangeably. However, probability and odds are not synonymous. Understanding the differences between probability vs. odds is essential, especially in fields like statistics, finance, and sports betting. Let's break down the definitions and key differences.

Definition of probability

The probability of an event indicates the mathematical likelihood that it will occur. In simpler terms, it is the fraction of all possible outcomes that result in your desired event.

Let's use an example to understand this clearly.

Example of probability

There are 12 face cards in a standard 52-card deck: the King, Queen, and Jack in each of the four suits.

Assume a friend shuffles the deck and asks you to draw a single card at random. You believe you can beat the odds, so you make a bet: if you do not draw a face card, you pay them $1. If you do, they pay you $5.

What is your winning probability?

Your probability of winning is the chance of drawing a face card out of all possible outcomes. Since there are 52 cards in the deck, there are 52 possible outcomes in total. Your desired event is drawing a face card. There are 12 potential outcomes for the desired event because the shuffled deck contains exactly 12 face cards.

To calculate probability, you express the total number of desired occurrences in relation to the total number of possible outcomes. That gives you 12/52. This is exactly how the winning probability is computed.

Definition of odds

Odds measure the likelihood of an event by comparing the number of desirable outcomes to the number of undesirable outcomes. In other words, odds represent the ratio between positive outcomes and unfavorable outcomes in a specific situation.

Let's use the previous card game example to understand this clearly.

Example of odds

In the example above, your favorable outcome is drawing a face card. As a result, there are 12 favorable outcomes. The number of unfavorable outcomes is calculated by subtracting the total number of favorable outcomes from the total number of all possible outcomes. You must subtract 12 from 52 because there are 52 outcomes in total.

No. of unfavorable outcomes = Total no. of outcomes - No. of favourable outcomes = 52 - 12 = 40

You can now use a ratio to express the total number of desired outcomes relative to the total number of undesired outcomes. This ratio is what we call "odds."

Probability calculation

Probability is calculated by dividing the number of desired outcomes by the total number of possible outcomes.

Probability = No. of desired outcomes / Total no. of outcomes

Let us now compute the winning probability for the previous example.

The winning probability = No. of face cards / Total no. of cards in the deck = 12 / 52 = 3 / 13

Next, we will compute the probability of losing. This is equivalent to estimating the probability of the complement of the desired event.

If the desired event is A, the complement event is denoted as Aᶜ or A¹. The probability of a complement event is calculated by deducting the probability of the desired event from 1.

P(Aᶜ) = 1 - P(A)

Let's calculate the probability of losing for our previous example.

We already calculated the winning probability as 3 / 13. Therefore:

Probability of losing = 1 - Winning Probability = 1 - 3 / 13 = 10 / 13

Odds calculation

Odds are computed by finding the lowest ratio between the number of desired outcomes and the number of undesired outcomes. This can also be determined by calculating the ratio between the probability of a desired outcome and the probability of an undesirable event.

There are two main types of odds calculations:

• the odds in favor,
• the odds against.

The odds in favor

The odds in favor represent the lowest ratio of the number of outcomes where the desired event happens to the number of outcomes where the desired event cannot happen. Let's say our desired event is A. The odds in favor of event A are calculated as follows:

Based on the number of outcomes:

The odds in favour of event A = n(A) : n(Aᶜ)

Based on probability:

The odds in favour of event A = P(A) : P(Aᶜ)

Let's calculate the odds in favor of winning in the card example given above.

1. Based on the number of outcomes

In the previous example, the desired event was drawing a face card.

No. of the desired outcomes = 12

No. of undesired outcomes = Total no. of outcomes - No. of desired outcomes = 52 - 12 = 40

Therefore,

The odds in favor = No. of desired outcomes / No. of undesired outcomes = 12 / 40 = 3 / 10

2. Based on probability

The desired event is drawing a face card.

The winning probability = No. of desired outcomes / Total no. of outcomes = 12 / 52 = 3 / 13

The probability of losing = 1 - The winning probability = 1 - 3 / 13 = 10 / 13

The odds in favour = The winning probability / The probability of losing = 3 / 13 : 10 / 13 = 3 : 10

The odds against

The odds against represent the lowest ratio of the number of outcomes where the desired event cannot happen to the number of outcomes where the desired event can happen. Assuming the desired event is A, the odds against event A are computed as follows:

Based on the number of outcomes:

The odds against event A = n(Aᶜ) : n(A)

Based on probability:

The odds against event A = P(Aᶜ) : P(A)

Let's calculate the odds against winning for the example given above.

1. Based on the number of outcomes

The desired event is drawing a face card.

No. of the desired outcomes = 12

No. of undesired outcomes = Total no. of outcomes - No. of desired outcomes = 52 - 12 = 40

Therefore,

The odds against winning = No. of undesired outcomes : No. of desired outcomes = 40 : 12 = 10 : 3

2. Based on probability

The desired event is drawing a face card.

The winning probability = No. of desired outcomes / Total no. of outcomes = 12 / 52 = 3 / 13

The probability of losing = 1 - The winning probability = 1 - 3 / 13 = 10 / 13

The odds against winning = The probability of losing : The winning probability = 10 / 13 : 3 / 13 = 10 : 3

Expression

Expression of probability

Probabilities can be easily expressed as a decimal, a percentage, a fraction, or a ratio.

In the previous example, we calculated the winning probability as a fraction:

• The winning probability = No. of desired outcomes / Total no. of outcomes = 12 / 52 = 3 / 13

We can express the winning probability as a decimal:

• The winning probability = No. of desired outcomes / Total no. of outcomes = 12 / 52 = 3 / 13 = 0.2308

The winning probability can also be expressed as a percentage:

• The winning probability = (No. of desired outcomes / Total no. of outcomes) × 100% = (12 / 52) × 100% = (3 / 13) × 100% = 23.08%

A ratio can be used to represent the probability of winning:

• The winning probability = No. of desired outcomes : Total no. of outcomes = 12 : 52 = 3 : 13

To summarize:

• The winning probability = 3 / 13 = 0.2308 = 23.08%

Expression of odds

Odds are usually expressed as a ratio reduced to its lowest terms.

According to our example:

• The odds in favor = No. of desired outcomes : No. of undesired outcomes = 12 : 40 = 3 : 10

• The odds against = No. of undesired outcomes : No. of desired outcomes = 40 : 12 = 10 : 3

The range

The range of probability

When an event is absolutely certain to happen, its probability is 1. When an event is impossible and will never happen, its probability is 0. As a result, the probability of any given event always falls strictly between 0 and 1. If the probability is expressed as a percentage, it will range between 0% and 100%.

The range of odds

The odds in favor are infinite when an event is certain to occur. Conversely, if the event is never going to happen, the odds are zero. Therefore, odds are mathematically represented as a number between 0 and infinity.

As per our example:

• The odds in favour = 3 : 10 = 0.3

• The odds against = 10 : 3 = 3.33

Converting odds to probability

As you have learned, odds represent the ratio between positive outcomes and unfavorable outcomes in a specific situation.

However, odds are not a direct expression of how likely an event is to happen. So, when you are given the odds, you may need to convert those odds to probability to understand the true likelihood of the event occurring. You can easily convert odds to probability using the following method.

If the favorable event is A, you know that:

n(S) = n(A) + n(Aᶜ)

Therefore,

$$P\left(A\right)=\frac{n\left(A\right)}{n\left(S\right)}=\frac{n\left(A\right)}{n\left(A\right)+n(A^c)}$$

Example of calculating the conversion of odds to probability

In our example:

• The odds in favor = 3 : 10

So,

• Probability of winning = No. of desired outcomes / (No. of desired outcomes + No. of undesired outcomes) = 3 / (3 + 10) = 3 / 13

Similarly:

• The odds against = 10 : 3

So,

• Probability of losing = No. of undesired outcomes / (No. of undesired outcomes + No. of desired outcomes) = 10 / (10 + 3) = 10 / 13

Converting odds to probability—or reducing odds to their lowest ratio—doesn't have to be a manual chore. An odds probability calculator can effortlessly help you convert winning odds to winning probability and reduce the odds for winning to their simplest terms. It will also reduce the odds against winning to their lowest ratio and seamlessly convert the odds against into the probability of losing.

To compute the answers for the previous example using an odds probability calculator, simply input 12 for A and 40 for B, choose "Odds are for winning," and hit calculate. You will get the exact same results if you enter 40 for A and 12 for B and select "The odds are against winning." Accurate answers will be ready in a split second.

The significance of odds

Odds have vital, real-world applications across multiple industries and fields.

In the scientific research sector, particularly regarding the transmission of diseases, odds are frequently utilized. To understand how an illness spreads and to develop effective public health responses, scientists use odds to compare the ratio of a population that develops a disease to the ratio of those who do not.

Financial experts rely on odds to evaluate risk-to-reward metrics, helping them determine whether a specific asset carries a significant risk or offers substantial potential gains before making crucial investment decisions.

Sports betting and gambling are other major sectors that heavily rely on odds. However, it's important to note that displayed betting odds rarely represent the true mathematical probability of an event happening. Bookmakers always bake a profit margin (the "vig" or "juice") into these odds. Because of this, the final payout to a winning bettor is always slightly lower than it would be if the odds perfectly reflected true probabilities.