Concepts in Probability #2: Happening vs Not Happening
Earlier this week, we did something hard: we looked at the difference between counting and computing. In the close of this two-parter on probability, I’d like to introduce you to another critical skill, which is based not on math, but on an ordinary fact of life: everything either happens, or it doesn’t. There’s no middle option. Either I’m going to eat some Skittles today, or I’m not. No halfsies.
This fact is very relevant for probability. In probability, the number “1″ is a guarantee, and it’s guaranteed that something either happens, or it doesn’t. As a result, the odds of something happening — plus the odds of it NOT happening — have to equal 1. In math terms:
P(It happens!) + P(It doesn’t happen!) = 1
Sometimes, when you’re asked to find the probability that something happens, it’s easier to compute the probability that it doesn’t happen, and subtract from 1. For convenience, let’s call this the “opposite rule.”
To be honest, I don’t think this is the first time you’ve been shown the “opposite rule.” Some well-intentioned math teacher somewhere probably already taught it to you. But if you still struggle with it, chances are it’s because you think the “opposite rule” is a trick.
It’s not a trick. From now on, I want you to regard the “opposite rule” as an option. And that option isn’t merely present on some probability problems: it’s available on ALL problems. Sometimes it’s a good option, and sometimes it’s not, and the skill you need to develop is distinguishing when is when. Let’s look at an example:
A fair coin is flipped six times. What’s the probability that fewer than five of the flips are tails?
To determine whether to approach this problem head-on or to use the “opposite rule,” consider the possible numbers of tails in each scenario:
“Fewer than five:” 0, 1, 2, 3, or 4
“NOT fewer than five:” 5 or 6
Notice how in the “not” scenario, there are half as many options as in the “happening” scenario! That’s how you can tell on Test Day that the “opposite rule” is a winner here. Find the probability that the event DOESN’T happen — in this case, that you WON’T flip fewer than five tails — and then subtract it from 1 to get the answer.
More counting is in order. We already counted last time how many ways there are to get exactly five tails out of six flips: there are 6 ways. Now, how many ways are there to get all six tails? This is a bit of a tricky question, but it’s one you’ll have to master asking yourself. There’s just one way: to get six tails, every flip needs to be a tail, and there’s only one way to do that.
Altogether, then, there are 6 + 1 = 7 ways to get what you want.
As we counted last time, there are 64 total ways the six flips could go down.
So the odds that you WON’T get what you want are 7 in 64. Don’t forget to subtract from 1 to get your answer: 64 – 7 equals a 57 in 64 chance that fewer than five of the flips will be tails.