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Author: Sophia Tutorial

This lesson will explain odds.

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What's Covered

In this tutorial, will discuss odds - odds in favor of an event or the odds against an event, by focusing on:

  1. Odds
  2. Converting Odds to Probability


Odds are often confused with probability. It's a different way to express likelihood that is different than probability. They're not the same. So when you say that there's a 1 in 5 probability, that's different than saying the odds in favor are 1 in 5.


On this spinner, there are three 1s, a 2, two 3s, and two 4s.

Suppose that 3 is a favorable outcome. Probability is the ratio of favorable outcomes to total outcomes. So there are two favorable outcomes out of eight total outcomes. And so it's 1/4 is the probability of 3.

Now we're going to contrast that with odds. The odds are the ratio of favorable outcomes to the unfavorable outcomes. So by contrast, the odds in favor of a 3, there are two favorable outcomes, whereas there are six unfavorable outcomes. And so the odds in favor of a 3 on the spinner are one to three.

This says the odds in favor of a 3 are one to three. Now that means that for every one favorable outcome, there are three unfavorable outcomes. Odds against are the unfavorable outcomes to the favorable outcomes, which means that there are three unfavorable to every one favorable.
Odds are usually expressed with a colon, rather than a fraction bar, although both are accepted. You should always reduce as if it was a fraction. Two to six versus one to three, one to three is preferred.

Odds can also be expressed against an event simply by reversing the numbers.

Odds against a 3 are three to one.

ExampleWhen odds are listed in the newspaper or a contest, typically what they're reporting to you are the odds against winning, so the odds against say, a horse winning a particular race.

Term to Know

    • Odds
    • A ratio relating the number of favorable outcomes to unfavorable outcomes (odds in favor) or vice-versa (odds against). Odds are usually expressed in simplest integer terms (eg 2:1, not 0.5:1 or 4:2).


Try It

The probability of getting heads on a coin is 1/2.

The probability of a red on a roulette wheel is 18 to 38.

The probability of rolling a 4 on a fair die is 1/6.

    1. Determine the odds in favor of heads and the odds against heads.
    2. Give me the odds in favor of red and odds against red.
    3. Give me the odds in favor of a 4 and the odds against 4.

On a coin, there's one favorable and one unfavorable outcome. On a roulette wheel, you might not have come up with this, 9 to 10 or 18 to 20.

But you do have to remember to reduce.

And then on the die, there is one favorable outcome and five unfavorable outcomes. For the odds against, you simply switch the numbers around.

Five unfavorable outcomes for every one time you get a 4.So use the marble jar. Determine these values for me.

Try It

Determine the following using this jar of marbles:

    • Odds in favor of red
    • Probability of green
    • Odds against blue
    • Probability of orange

What you should have come up with is this:

    • Odds in favor of red = 7:11
    • P(green) = 5/18
    • Odds against blue = 7:2
    • P(orange) = 1/18

There are 18 marbles in the jar. 7 of them are red. And 11 of them are not red.The probability of green, there are 5 greens out of 18 total.

The odds against blue, there are 4 blues and 14 that aren't. You should started by saying 14 unfavorable to 4 favorable, and reduced it. And the probability of orange, there's just the 1 lone orange marble out of 18.

So go ahead one more time, one more practice.

    • P (red)
    • Odds in favor of green
    • P (blue)
    • Odds against orange

And you don't necessarily have to use the marble jar. Try and use your answers from the previous versions of the problem.

What you should have come up with this time were these numbers:

    • P (red) = 7/18
    • Odds in favor of green = 5:13
    • P (blue) = 2/9
    • Odds against orange = 17:1

For red, 7 out of 18 marbles. You could realize this by saying there are 7 favorable outcomes and 11 unfavorable outcomes, therefore there are 18 total outcomes, 7 of which are favorable.

Using this probability, you can say there are 5 favorable outcomes, which are green. And the remainder of the 18, which are 13, are not favorable, as in not green.

The probability of blue is 2, which was the reduced value of blue here, out of 9, which is the unfavorable and favorable added together. And then finally the odds against orange are 17 to 1, 17 not orange and 1 orange.


Odds are not the same as probability. The two terms should not be used interchangeably, although in real life you will hear it often. The odds in favor of an event are the favorable to unfavorable outcome ratio. The odds against an event are the unfavorable to favorable outcome ratio.

Odds can be converted to probability by evaluating favorable to unfavorable outcomes.

Good luck!

Source: This work is adapted from Sophia author jonathan osters.

Terms to Know

A ratio relating the number of favorable outcomes to unfavorable outcomes (odds in favor) or vice-versa (odds against). Odds are usually expressed in simplest integer terms (eg 2:1, not 0.5:1 or 4:2).