Probability, Part 6, The Problem with Probability

In my last few posts, I’ve talked about probability and how to calculated a basic probability:

Probability = \[\frac{{\mathrm{Number}}\hspace{0.33em}{\mathrm{of}}\hspace{0.33em}{\mathrm{favorable}}\hspace{0.33em}{\mathrm{outcomes}}}{{\mathrm{Total}}\hspace{0.33em}{\mathrm{number}}\hspace{0.33em}{\mathrm{of}}\hspace{0.33em}{\mathrm{possible}}\hspace{0.33em}{\mathrm{outcome}}{s}}\]

This formula is simple if you know the number of favourable outcomes and the number of possible outcomes. This works well if asking questions like what is the probability of rolling  a 7 with a pair of dice. To calculate the number of total outcomes, there are 6 possible ways a single die can be thrown, and for each of these, the other die can have 6 possible value. So the total will be 6 × 6 or 36. This illustrates the multiplication rule for counting things:

If there are m ways for one thing to occur and n ways for a second thing to occur, then there are m × n ways to do both.

Manually counting the ways to get a 7 where the first number is from die 1 and the second from die 2 gives:

1 + 6, 2 + 5, 3 + 4, 4 + 3, 5 + 2, and 6 + 1. Six ways

So the probability of rolling a 7 is 6/36 or 1/6.

Now what if I asked what is the probability of getting four aces in a 5-card poker hand? How do you even begin to count the number of possible poker hands? There are two ways to count large possibilities like this: combinations and permutations.

A combination is the number of ways where a collection of objects can be arranged where you are not concerned with order. For example, in the card example, a hand of 2, 3, 4, 5, and 6 of hearts would be the same as a 6, 5, 4, 3, an 2 of hearts and you would only want to count these two possibilities as one along with any other arrangement of these five cards. A permutation is where order does count and these two card combinations would be counted as two permutations.

In our card example, order doesn’t count, so we want the number of  combinations of taking 52 cards, 5 at a time. Fortunately, there is a formula and notation used to simplify this. Before I present this, there is another math operation that needs to be explained: factorials.

You may have a calculator with a “!” symbol or “x!” on one of the keys. This is a factorial operation. A factorial is successively multiplying an integer by one less for each factor. For example, 5! = 5 × 4 × 3 × 2 ×1 = 120. Factorials get large very quickly. For example, 30! is 265252859812191058636308480000000. To make the formulas using factorials consistent, a special definition 0! = 1 is made.

So the notation for the number of r combinations of n objects is \[
\] or more commonly C(nr). So in our case, we want to calculate C(52,5), this is the number of possible 5-card combinations out of a deck of 52 cards. The general formula for combinations is


In our poker hand example, the number of possible poker hands is


Now before you go off and calculate this, remember how large factorials can get? Many calculators cannot keep the number of digits necessary to accurately store very large numbers and the accuracy of the calculation will be poor. So when dealing with combination and permutation formulas, it is always best to simplify before calculating the answer. See if you can see where we can simplify the expression on the right side. I will continue this example in my next post.