Last time, we looked at some overarching laws of probability. We discovered that luck always balances out in the long run and that dice don’t hate me really!

We also discovered that some risks are much riskier than others. How do you know when to take a risk and when to be cautious? How do you actually calculate the odds of succeeding when rolling the dice? This time we’re going to get into the nitty gritty of calculating probabilities.

Warning! This post contains actual Maths! Not too much hopefully, as the basic principles really aren’t that complicated. Let’s go back to school…

### Conventions

Probabilities can be expressed as percentages (eg. there’s a 30% chance of rain today) or as odds (eg. there’s a 1 in 4 chance of drawing a heart from a pack of cards) or as fractions (these are essentially the same as odds, so 1 in 4 could be written as ).

Mathematicians prefer fractions as the rules for adding and multiplying fractions are unambiguous. There is one other common expression though, which is when the chances of something happening is 50-50. This means a 50% chance of happening and a 50% chance of not happening (eg. flipping a coin and getting heads).

### Fractions

Yes, I’m very sorry, but you do need to know some basic fraction arithmetic. Don’t worry though, I’ll keep it simple.

To multiply fractions, you just multiply the numbers on the top and then multiply the numbers on the bottom so because 1 x 1 = 1 on the top and 2 x 2 = 4 on the bottom. Easy peasy.

Adding fractions is a little harder, but as long as the numbers on the bottom are the same, you just add the numbers on the top and *don’t change the number on the bottom!* That’s a very common mistake so watch out for that. not ! Notice that would be the same as (think slices of pizza: two quarters of a pizza is the same as one half of a pizza).

We won’t worry about any other fraction operations. Those should be enough for most calculations.

### Calculating Probability

The essential definition of probability refers to the likelihood of something happening (eg. rolling a 6 on a d6). It also sounds more complicated than it really is so I’m going to just illustrate it with an example.

If you want to find the probability of rolling an even number on a d6, you work out how many outcomes meet your requirement (in this case 2, 4 and 6 are all possible so that’s 3 outcomes) and then divide that by the total number of possible outcomes (ie. 6). So 3 divided by 6 gives you or .

If you’re rolling two dice and want to know the chances of rolling a particular total (eg. 10), you have to consider all the possible outcomes. It’s easiest to do this with a table:

To obtain 10, you could roll:

- 6 on Dice 1 and 4 on Dice 2;
- 5 on both dice; or
- 4 on Dice 1 and 6 on Dice 2.

You can see the three possibilities really easily in the table (highlighted yellow). So the chances of rolling 10 would be 3 in 36 (there are 6×6=36 outcomes altogether in the table), which could also be expressed as .

To find out the chances of rolling 10 **or more**, you just count up all the 10s, 11s and 12s. In this case there are 6 possibilities (three ways of getting 10, two ways of getting 11 and one way of getting 12). This gives , which you could express as 1 in 6. So if you roll two dice, this means you would expect to roll 10 or more one sixth of the time.

As you can see, the trick to working out many probabilities is just counting up the number of possibilities correctly.

When combining probabilities, there are two helpful rules of thumb that I like to use. They are called ‘The AND Law’ and ‘The OR Law’

### The AND Law ()

This says that if you want to know the chance of one thing AND another thing happening (eg. rolling a 6 on a d6 AND flipping a coin and getting heads), then you have to **multiply** the two probabilities.

So the chance of rolling a 6 AND getting heads would be . The chance of rolling two 6s in a row (rolling a 6 AND then rolling another 6) would be .

### The OR Law (+)

This says that if you want to know the chance of one thing OR another thing happening (eg. rolling an odd number OR rolling a 6 on a d6), then you have to **add** the two probabilities.

So the chance of rolling an odd number OR a 6 would be , which would be the same as .

Probability can obviously get more complicated than this, but hopefully that gives you a starting point towards calculating the odds of success whenever luck is involved. Remember to compare your odds with what you stand to gain/lose when rolling the dice! And good luck!

I’m sure you know it, but since the post is for people with little background in probabilities, I would point out that combining the probabilities of two events with the “AND” and “OR” only works if the two events are independent (for example, the probability of getting a 6 or an even number is 1/2 and not 1/6+1/2)…