A Probability Primer

If you've ever played a game with dice, you've done some probability. Most people can estimate probabilities instinctively; our capacity to think abstractly and analytically is what separates humans from most animals. Still, probability estimation is a bit inexact, subject to personal experience and emotion. How can we make it more precise and reliable?

With the power of mathematics!

What is probability, and how do we calculate it?

Probability is a ratio, comparing favorable outcomes to all outcomes. To find the probability of some event, we just have to find out how many ways we get what we want (the favorable outcome) and how many ways things could happen in total. Let's say we want to know the likelihood of rolling a 2 on a six-sided die. We want a 2, and there is one way to get it. The possible outcomes are 1, 2, 3, 4, 5, and 6. There are six possible outcomes, and one of them is what we want. Therefore, our ratio is 1 to 6, or 1/6.

The probability of rolling a 2 on a six-sided die is 1/6, or about 17%. We can make this into an equation:

(1) P(A) = (# of outcomes that make A)/(all the outcomes)

Next, let's tackle two dice. What is the probability of rolling two dice, and getting snake-eyes? That means we need to roll a 1 on both dice. The rule is, for events A and B:

(2) P(A and B) = P(A) * P(B)

In words: the probability of rolling a 1 on the first die and rolling a 1 on the second die is the product of the two probabilities. Figure out how likely each of them is, then multiply. So let's do the example: the probability of rolling a 1 on the first die (event "A") is 1/6. The probability of rolling a 1 on the second die (event "B") is 1/6. Therefore: P(A and B) = 1/6 * 1/6 = 1/36, or about 2.8%.

This rule comes with a caveat: events A and B must be independent. In other words, the dice don't care what the other die does. Rolling one die does not affect the other. If the events are not independent (such as the probability of me stubbing my toe and the probability of me swearing), then we have to use a different formula.

Next, let's consider the probability of rolling a 3 and a 2 on my dice. Well, I can roll this two ways: I can roll a 2 on the first die and a 3 on the second, or a 3 on the first and a 2 on the second. Either works. I need to introduce a new rule here:

(3) P(A or B) = P(A) + P(B)

In words: the probability of either event A or event B occurring is the sum of the two probabilities. Just find the probability of each one, then add them.

Now let's break this down. The first way to get "a 3 and a 2" is to roll a 2, then a 3. Probability? First, roll a 2; 1/6th chance. Then, also roll a 3; 1/6th chance. Rule #2 states that I should multiply these: 1/6 * 1/6 = 1/36. Therefore, the probability of rolling first a 2, then a 3 is 1/36. Next, the other option: roll a 3, then a 2. Follow the same logic and we find that this is also a 1/36th chance.

Now, use rule #3, and sum them. 1/36 + 1/36 = 2/36 or about 5.6%.

Hold on: how did I know what rules to use when? I've been using "and" and "or" in my formulas in part because their appearance in the question tells you which formula to use. I said: "roll a 2 on the first die and a 3 on the second, or a 3 on the first and a two on the second."

If I wanted to write this more symbolically, I could say: (2 and 3) or (3 and 2).

If two events must happen together, we're looking for their combined probability, that's an "and." If we have two events, and either of them is fine, then that's an "or". So, we do need a 3 and a 2, but either "3 then 2" or "2 then 3" satisfies.

These three rules will set you up well for computing basic probabilities. There are a few more variations on these rules. First, we have the "not." What is the probability of rolling a die and not getting 3? Well, it's the combined probabilities of everything except 3. So we take everything (probability: 1) and subtract out the probability of 3, which is 1/6. Result: 5/6, or about 83%. (The ! below is shorthand for "not.")

(4) P(!B) = 1 - P(B)

The second variation is so simple it barely needs explanation. Basically, we've been talking about probabilities with two events, but we can do it with three or more events quite easily by extending rules #2 and #3.

(2a) P(A and B and C and...) = P(A) * P(B) * P(C) * ...

(3a) P(A or B or C or...) = P(A) + (P(B) + P(C) + ...

The thorny part of probability is not doing the math, but instead clearly stating what is being asked. Take that "roll a 2 and a 3" problem. Does it matter which die is which? Did I mean: "roll a 2 on the first die, then roll a 3 on the second die" or did I mean "roll a 2 and a 3, in whatever order"? There's only one way to get the first scenario, but two ways to get the second. This is one taste of how whether order matters or not changes the realm of answers and the probability of a given event. Another issue is, when counting outcomes, not to miss any or count some twice. Yet another is when the various outcomes are not all equally likely (playing with loaded dice, for example.)

All these can make the study of probability quite interesting indeed! But more advanced math is for another time. If you're curious, feel free to check out Dice and Gaming for some more die-based probability fun.