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How are true odds calculated?

The answer is the total number of outcomes. Probability can be expressed as 9/30 = 3/10 = 30% - the number of favorable outcomes over the number of total possible outcomes. A simple formula for calculating odds from probability is O = P / (1 - P). A formula for calculating probability from odds is P = O / (O + 1).

wikihow.com - How to Calculate Odds: 11 Steps (with Pictures) - wikiHow

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Let's do an example problem. To calculate the odds of rolling two dice with a sum of four (for instance, a 1 and a 3), begin by calculating the total number of outcomes. Each individual dice has six outcomes. Take the number of outcomes for each die to the power of the number of dice: 6(number of sides on each die) 2(number of dice) = 36 possible outcomes. Next, find the number of ways you can make four with two dice: you can roll a 1 and a 3, a 2 and a 2, or a 3 and a 1 - three ways. So the odds of rolling a combined "four" with two dice are 3 : (36-3) = 3 : 33 = 1 : 11 Next, find the number of ways you can make four with two dice: you can roll a 1 and a 3, a 2 and a 2, or a 3 and a 1 - three ways. So the odds of rolling a combined "four" with two dice are Odds change exponentially based on the number of events occurring simultaneously. Your odds of rolling a "yahtzee" (five dice that are all the same number) in one roll are very slim - 6 : 65 - 6 = 6 : 7770 = 1 : 1295! If we roll one die, it's equally likely that we'll get any of the numbers 1 - 6. However, if we rolldice and add their numbers together, though there's a chance we'll get anything from 2 to 12, not every outcome is equally likely. There's only one way to make 2 - by rolling two 1's - and there's only one way to make 12 - by rolling two 6's. By contrast, there are many ways to make a seven. For instance, you could roll a 1 and a 6, a 2 and a 5, a 3 and a 4, and so on. In this case, the odds for each sum should reflect the fact that some outcomes are more likely than others.

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