# 2.1.1.1 - Risk and Odds

You may have heard the terms risk and odds before. They are both ways to communicate the likelihood of an event.

Risk and odds are often confused with one another. The formulas for computing risk and odds are different and their interpretations are different.

Risk

In statistics, the word risk communicates the likelihood of an event occurring. This is synonymous with probability or proportion (i.e., the formulas are the same).

Risk
The probability that an event will occur. It may be written as a decimal, a fraction, or a percent.
Risk
$$Risk= \dfrac{number \;with \;the\; outcome}{total\;number\;of\;outcomes}$$

## Example: Asthma Risk Section

$$60$$ out of $$1000$$ teens have asthma.

$$risk=\dfrac{60}{1000}=0.06$$

This means that $$6\%$$ of teens experience asthma.

## Example: Flu Risk Section

$$45$$ out of $$100$$ children get the flu each year.

$$risk=\dfrac{45}{100}=0.45$$ or $$45\%$$

## Odds Section

Odds
Express risk by comparing the likelihood of an event happening to the likelihood it does not happen.
Odds

$$odds = \dfrac {number \;with \;the\; outcome}{number \;without \;the \;outcome}$$

OR

$$odds=\dfrac{risk}{1-risk}$$

We often interpret odds in relation to the value of 1. For example, if the odds of a game are in favor of the house 2 to 1, that means for every 2 games the house wins it will lose 1.

## Example: Passing Odds Section

In one large class, 850 students passed an exam while 150 students failed. Because we have the raw counts, we can use the first odds formula.

$$odds=\dfrac {number \;with \;the\; outcome}{number \;without \;the \;outcome}=\dfrac{850}{150}=5.667$$

The odds of passing were 5.667 to 1. In other words, for every 5.667 students who passed the exam there was 1 who failed.

## Example: Flu Odds Section

The risk of a child getting the flu is $$45\%$$ which can also be written as $$0.45$$. Because we have the risk, we can use the second odds formula.

$$odds=\dfrac{risk}{1-risk}=\dfrac{0.45}{1-0.45}=\dfrac{0.45}{0.55}=0.818$$

The odds of a child getting the flu is $$0.818$$ to $$1$$.