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Author: Ryan Backman

Relate probability to odds in a given situation.

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Hi. This tutorial covers odds. So let's just start with a statement here. So the odds against rolling snake eyes is 35 to 1. OK? So when we're talking about snake eyes, we're talking about rolling two dice, and snake eyes are the double 1's. OK?

So we want to first think about probabilities. So how are odds related to probability? So if we look at the probability-- so this is probability notation-- the probability of snake. So we know that there are six possible outcomes on each die, so there are 36 total outcomes if you roll a pair of dice. And since there's only one way to get a 1 and a 1, the probability of rolling snake eyes is 1 out of 36.

So what we're going to figure out is how do these odds relate to the probability? OK. So just to define odds, odds is an expression of relative probabilities. OK? So we all have odds in favor and odds against. So if the probability of an event is A over B, then the odds in favor are A to B minus A, and the odds against are B minus A to A. OK?

So now, let's actually figure out how to calculate those odds for our snake eyes example. OK. So if the probability of rolling snake eyes was 1 out of 36, remember that this is A and this is B. OK? So the odds in favor-- so if we're doing odds in favor, that's A to B minus A. OK? So A is 1, and B minus A is 36 minus 1. OK?

So the odds here are 1 to 35. So the odds in favor of you rolling snake eyes is 1 to 35. OK? Now, the odds against are always the odds in favor flipped around. So in this case, it's B minus A to A. So this ends up being 35 to 1 odds against snake eyes.

OK. A couple of things about odds. So let's say, you ended up with odds of 14 to 4. OK? Generally, we reduce odds like we reduce fractions. So if we reduce this, we could divide both of these two by 2. So these odds are the same as the odds of 7 to 2. OK?

So this is probably the best way of writing those odds. You don't want to reduce again. You don't want to write it as 3.5 to 1, so dividing each of those by 2. OK? Simply keep it-- the lowest terms here would be 7 to 2. OK? So that's just a way that odds are typically represented.

OK. Now, if we go the opposite way here, and we start with the odds, we want to figure out, well, what's the probability? OK? So if the odds for an event are A to B, the probability can be determined as A over A plus B. OK. So if we're looking at the odds against rolling two dice with a sum of 7 are 5 to 1. OK?

Now, this is the formula for odds for an event, or odds in favor of an event. OK? You'll see it both ways. You see it as odds for or odds in favor. OK? Those are both used interchangeably.

So if we know that the odds against are this, we know that the odds in favor is going to equal 1 to 5. OK? So what that means is that the 1 is A and the 5 is B. OK? So now, if we set this up, it's going to be A over A plus B. OK?

So A is 1, so this is going to be 1 over A plus B which is 1 plus 5. OK? So this ends up being 1 over 1 plus 5 is 6, so 1/6. OK? So the probability of rolling a sum of 7 is 1 over 6. OK?

Now, if we think about what we already know about dice, there are six ways of getting a sum of 7. OK? So if we take those six ways, and we divide by the total number of possible two-dice combinations, we end up with 6 out of 36 which reduces to 1/6 there. So that probability does check out, when we start with odds and convert to a probability. So that has been the tutorial on odds. Thanks for watching.

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).