How to Identify Common Probability Misconceptions

Instructor: Matthew Kantrowitz

Matt has a master's degree in political science and has taught at the college level

In this lesson, you will learn about a number of common misconceptions and how to avoid them. When will your luck change? What does it mean for a player to be on a hot streak? Here are a few and how to avoid them.

How to Identify Common Probability Misconceptions

You're watching your friend Mickey play a dice game. He rolls two dice, and if he rolls a seven, he wins. The last five rolls, he's gotten a two, a four, a five, an eight, and a twelve. It's starting to get grim, and from the way that big guy with the scar is looking at him, you're starting to get a little worried for Mickey. You tell him maybe it's time to pack it in, but he's somehow become more confident than before he's going to get a seven this time. ''Don't you see?'' He says, ''my luck is bound to change soon! That's what the probability says!'' You aren't so sure he's right. But then, sure enough, he rolls a seven. ''I knew it!'' he shouts. Was he right all along?

The Gambler's Fallacy

Even though Mickey won this time, you were right to think he was crazy to be so sure his luck would change. But first, let's see why he was so sure his luck would change. First, Mickey realized there are six ways to roll a seven with two dice: (1,6) (2,5) (3,4) (4,3) (5,2) and (6,1). Then, he reasoned there are 6 sides x 6 sides = 36 possible combinations of numbers. That means the probability of Mickey rolling a seven is 6/36, or 1/6. Mickey reasoned that if he had already rolled five times and not hit a seven, the sixth time must be the charm. Although it actually sounds pretty reasonable, Mickey has committed the gambler's fallacy. Even though he won, his luck never actually changed.

It All Depends (or Does It?)

To know why, it is necessary to understand the difference between dependent and independent events. For a 'dependent' event, its probability depends on the events leading up to it. For example, the probability of drawing the Ace of Spades from a deck of cards is 1/52. If you don't draw it the first time and discard whatever you drew instead, there are now only 51 cards left, so the probability goes up to 1/51. If you keep drawing cards and discarding them, the probability of drawing the ace goes up until you get it. Rolling dice is an 'independent' event. No matter how many times you roll, and no matter what you get, the probability of rolling a seven with two dice is 1/6. Whether you throw ten sevens in a row or don't get a seven for 20 straight rolls, the probability is always 1/6.

Recency Bias

The opposite of the gambler's fallacy is called recency bias. Human beings have a unique tendency to subliminally put more weight on events that happened in the recent past. If events are independent, it doesn't matter how many times in a row you flip heads with a fair coin. The next flip is a 50/50 shot to come up heads again.

Hot Hands and Small Samples

There are also misconceptions around sample size. Imagine you and Mickey are playing basketball one on one. He usually makes 80% of his free throws, but he's having an off day. He's only made 1 out of 5, or 20%. If you foul him again, what is the chance he'll make his next free throw? You might think it's 20%, but five is not a big enough sample size to know if Mickey really is worse than you thought. Your best guess is Mickey has an 80% chance to make his next shot. So you probably shouldn't foul him again.

What are the odds?

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