That Common Misconception About Probability

Dependent vs Independent Events

A couple posts ago I began talking about probabilities. I want to spend some more time on the topic because it’s one of those concepts that can be obviously easy one minute and down right confusing the next.

Dependency: The Basics

Whenever you approach a probability problem involving more than one event begin by asking yourself whether the events are dependent or independent, i.e. does one event alter the probability of the other event? Does the first event happening change the system?

(Drawing multiple cards from a deck without replacement is a classic example of a dependent event. When we draw the first card the system changes from 52 possible options to 51 before the second event happens.)

In real life, humans have a tendency to confuse dependency all the time. For instance, you flip a fair coin. It lands heads 10 times in a row. You are certain the next flip is bound to be tails…right?

What’s the chances of the next coin being heads again?

It’s still 1/2. Flipping a coin is an independent event. In other words, the outcome of the next flip is uninfluenced by what has happened previously. It’s as if it were the first time you ever flipped the coin. The probability is unaltered.

Why does it “feel” like it should be tails then?

We tend to think about the events together, instead of individually. Although the probability of tossing heads remains the same for each toss, the combined chance of tossing 11 heads in a row is small.

Let’s calculate it.

We learned in the previous probability lesson that when we string multiple events together and want them all to occur (the “and” scenario) we must multiply their probabilities together.

Since each coin toss has a probability of heads equal to 1/2, I simply need to multiply together 1/2 eleven times.

Probability of flipping eleven heads in a row

That’s a 0.05% chance of flipping eleven heads in a row! But before you exhale of sigh of relief and say, “See, I knew it!”, let’s calculate the chance of getting a tail instead on the eleventh toss.

Probability of flipping 10 heads followed by 1 tail.

Yes, that’s right. It’s equally likely to flip ten heads followed by a tail as it is to flip eleven heads in a row. In fact, because the individual probability of flipping heads is the same as the probability of flipping tails, each arrangement of 11 coin tosses will result in the same likelihood of 0.0005. Every arrangement is equally likely.

So… why does it “FEEL” improbable to flip ten heads in a row??

The common misconception isn’t caused by probability, but by a misunderstanding of combinatorics and permutations.

So far we’ve established that:

  • The probability of flipping heads or tails is equally likely each individual toss: P(H) = P(T) = 1/2.
  • Each unique arrangement (permutation) of possible coin tosses is equally likely.

So what gives?

Understanding Sample Space

The sample space is simply a listing of all the possible outcome arrangements (permutations). Since the sample space for 11 consecutive coin tosses is rather large, let’s examine a simpler case instead.

The sample space for four coin tosses is:

Intuitively, we might think: It’s more likely to flip 2 heads and 2 tails out of four tosses than all heads or all tails.

And this is true. Let’s do the math.

We know that each of the sixteen permutation’s are equally likely because P(H) = P(T) = 1/2. So each permutation has an equal probability of:

Note: Because the sample space represents all possible outcomes, the sum of all probabilities always equals 1 (E.g. 0.0625 • 16 = 1).

Looking at our sample space, how many different coin toss permutations result in any combination 0f 2 heads and 2 tails?

Sample space for four coin tosses

Notice that 6 out of 16 possible outcomes result in a two head/two tail combination. Because any one of those six permutations meets our criteria, this is considered an “or” scenario — so add the probabilities together (or simply multiply by 6 since they’re all the same).

That’s a 37.5% chance of tossing a combination of 2 heads and 2 tails, which is far greater than the probability of tossing all heads or all tails (which remains 6.25% each since there is only one permutation of each in our sample space).

This is why we intuitively know it is more likely to flip an equal number of heads and tails, than to flip all of one kind, while still maintaining the fact that the events are independent with individual tosses and permutations equally likely to occur.

Permutations & Combinations

Much of successfully computing probabilities is understanding permutations and combinations thoroughly. So that’s where we are headed next!

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