Tag Archives: probabilities

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Casino Probabilities : A must read intro

 

 

The casino roulette outcome was red 8 times in a row! What next?

Is black more likely on the next spin? No, it is not. Both red and black are equally likely. If you thought otherwise then the casinos love you, and you need to read this article right now.

In this article we’ll show here is why past events have no influence over future events. To understand this you need to know just a teeny tiny bit of math, and just one term, probability. Probability describes how likely it is that something will happen. There are three ways to refer to it: by fraction, by decimal, or by percentage. For example, say there are four cards, face-down, and you get to pick one. Three of them are aces. What are your chances of picking an ace? You have three chances out of four to get an ace. We can express this in any of these ways:

  • 3/4    (fraction)
  • 0.75   (decimal)
  • 75%   (percentage)

Each of these is just a different way of talking about the same thing. Notice that they’re pretty easy to convert, too. If you punch 3 divided by 4 into a calculator you get 0.75. And to convert a decimal to a percentage all you have to do is move the decimal two spaces to the right and add the percent sign. 0.75 is the same as 75%. What could be easier?

Okay, so now that we know how to refer to probabilities, let’s look at what they mean. Something that definitely will happen has a probability of 1 (or 1/1, or 100%, if you prefer). There’s a 100% chance the sun will come up tomorrow. Well, it’s not really a “chance” since it definitely will happen, but you get the idea. In our example of four cards, if all four were aces then your chances of picking an ace would be 4/4 = 1, it would definitely happen.

Something that definitely will not happen has a probability of 0. And in between 0 and 1 (or 0% to 100%) are all the things that could happen.

Your chances of winning some bet or series of bets might be 22%, 39%, 57%, or 83%. The higher the number, the more likely it will happen. Events over 50% will probably happen, events under 50% will probably not happen.

So far so good. So now let’s look at probability when an event happens many times, like flipping a coin over and over. The probability of getting heads on one flip is 1 out of 2 — one way to win out of two possible outcomes. We can call that 1/2 or 0.50 or 50%. But what are the chances of flipping the coin twice and getting heads both times? To figure this we multiply by the probability of each event:

 

First Flip
Second Flip
Probability
x
=

Of course, another way to express this is 50% x 50% = 25%.

Okay, so what are the chances of getting ten heads in a row?

 

x
x
x
x
x
x
x
x
x
=

Not very likely, of course.

So here’s where the gambler’s fallacy comes in: Say you’ve tossed the coin nine times and amazingly, you got nine heads. You figure that the next toss will be tails, because the probability of getting ten heads in a row is one in 1024, which is unlikely to happen!

The problem with this reasoning is that you’re not looking at the chances of getting ten heads in a row, you’re looking at the chances of getting one heads in a row. The heads that already happened no longer have a 50% chance of happening, they already happened, so their probability is 1. When you flip again the odds for that flip will be 50-50, same as it ever was.

Let’s introduce our hero, Mr. P, who will always be looking to the future to see what’s going to happen. He’s about to make ten coin flips, hoping to get ten heads. Here’s his outlook:

 

x
x
x
x
x
x
x
x
x
=

And here’s Mr. P. after flipping nine heads in a row, getting ready to make his tenth flip:

 

x
x
x
x
x
x
x
x
x
=

Now you’re saying, Hey, wait! How come all the 1/2’s turned into 1’s? The answer is that they’re no longer unknowns. Before you flip a coin you don’t know what’s going to happen so you have 50-50 odds. But after you flip the coin you definitely know what happened! After you flip a coin, the probability that you got a result is 1. You definitely flipped the coin. Definitely, definitely. So after you’ve flipped nine heads, the probability of flipping a tenth head is 1x1x1x1x1x1x1x1x1x 1/2 = 1/2.

Let’s have another look at Mr. P:

 

x
x
x
x
x
x
x
x
x
=

Notice that it doesn’t matter where on the table you stick him, the chances of his next flip being heads is always 1/2. Wherever he is, it doesn’t matter what happened before, his chances on his next toss are always 1 in 2.

How could it be otherwise? When you flip a coin you will get one result out of two possible outcomes. That’s 1 in 2, or 1/2. Why and how could those numbers change just because you got a bunch of heads or tails already? They couldn’t. The coin has no memory, it neither knows nor cares what was flipped before. If it’s a 1-out-of-2 coin, it will always be a 1-out-of-2 coin.

Still not convinced? Then here’s another way to think about it. Let’s say someone hands you a coin and asks, “What are the chances of flipping heads?” Without hesitation you’d probably say 1 out of 2? But wait a minute — if it were true that heads were more likely if tails has just come up a bunch of times, then why did you answer “1 in 2” right away when asked about the chances of getting heads? Why didn’t you say, “Well, you have to tell me whether tails has been coming up a lot before I can tell you whether heads has a fair shot or not.”? It’s simple: You didn’t ask about the previous flips because intuitively you know they’re unimportant. If someone hands you a coin, the chances of getting heads are 1 in 2, regardless of what happened before.

Would it really be the case that you answered “1 in 2,” and then your friend said, “Oh, I forgot to tell you, tails has just come up nine times in a row.” Would you now suddenly change your answer and say that heads is more likely? I hope not.

One last example: Let’s say your friend slides two quarters towards you across the table. He tells you that the first coin has been flipping normally, but the second quarter has just had nine tails in a row. Would you now believe that the chances of getting heads on the first coin are even but the chances of getting heads on the second coin are greater? Given two identical coins, could you really believe that one would be more likely to flip heads than the other? I hope not!

The same concept applies to roulette. An American roulette wheel has 18 red spots, 18 black spots, and 2 green spots. The chances of getting red on any one spin are 18/38. If you just saw nine reds in a row, what is the likelihood of getting black on the next spin?

18/38, same as it ever was.