What Are the Odds?

As we write this, Pat is just back from a trip to Las Vegas. The action at the craps tables reminded her of Fritz Leiber's Nebula-Award-winning story, "Gonna Roll the Bones," in which Joe Slattermill rolls dice with Death. Playing the slots called to mind Harlan Ellison's "Pretty Maggie Moneyeyes," in which a down-and-out gambler wins a fortune with the aid of a woman whose soul is trapped in a slot machine. And Pats knowledge of probability led her to contemplate the biggest fantasy of all—the fantasy that you can win in Vegas.

Examining that last fantasy brings us to this column and an analysis of probability, an area of math that most people don't really understand. Calculating probability can be tough. To calculate the odds that something will happen, you have to count the ways that something can happen and then count the ways it cannot happen. And as Paul is fond of saying, counting is difficult.

"Wait a minute," you say. "Back in first grade, counting was easy. There were five apples, two oranges. Nothing to it." Yet when you get around to reading The House at Pooh Corner, you discover otherwise. Christopher Robin and Pooh visit "an enchanted place at the very top of the Forest called Galleons Lap, which is sixty-something trees in a circle; and Christopher Robin knew that it was enchanted because nobody had ever been able to count whether it was sixty-three or sixty-four, not even when he tied a piece of string round each tree after he had counted it." (You see, to count the trees you have to keep track of large numbers and figure out which ones are trees and which ones are bushes. . . . and that's hard.)

In this column, we'll tell you a bit about probability and tell you the best strategy for winning on "Let's Make a Deal." We'll show you how to calculate the probability of certain events and explain why we don't play the California Lottery—even when the jackpot is $40 million.

We'll start our discussion of probability and gambling with an intriguing question that mathematicians have dubbed the Monty Hall problem. It's named after the host of the popular game show "Let's Make a Deal" and is based on an idealized version of a moment in that show.

Let's say you're a contestant on this version of "Let's Make a Deal." You dressed up in a costume that was sufficiently goofy to get Monty to pick you from the other lunatics in the audience. You trade Monty whatever it is you brought to trade—a swizzle stick in the shape of a hula girl, let's say—and he offers you a choice of three doors. There's something great—let's say a trip to Hawaii—behind one of the doors.

With much prompting from the audience, you choose Door Number One. In this version of the show, Monty always opens one of the doors that you didn't pick, and it never has the big prize.

So let's say Monty opens Door Number Three, revealing five thousand rolls of toilet paper. At this point, Monty asks you if you want to change your choice: do you want to stay with Door Number One or switch to Door Number 2?

Visions of luaus dance in your head and you freeze. What should you do? Should you switch or would you be better off staying with your original choice? Or does it matter? Are the odds the same either way?

And for those who know probability, the best choice is very clear. But most people don't see it that way. While we talk about probability, think about your choice of doors and whether or not you want to switch.

Probability is a way to measure how likely something is to happen. When you flip an ordinary coin, you have an equal chance of getting heads or getting tails.

If you flipped a coin one hundred times, it's likely that you'd get about as many heads as you'd get tails. But probability doesn't guarantee equal numbers of heads and tails. Probability doesn't tell you what will happen. It just tells you how likely something is to happen.

You win a coin toss by correctly calling whether heads or tails will face upward when the coin comes to rest. You figure out the probability that you will win by counting both the total number of possible outcomes and the number of those outcomes in which you win. In this case, there are two possible outcomes: heads and tails. One of those outcomes means you win. So the probability that you will win is 1 out of 2—which you can write as 1/2.

Looks like a fraction, doesn't it? It is. Probability is a number from 0 to 1 that measures the likelihood that an event will occur. A probability of 0 means that an event will never happen. A probability of 1 means that the event will certainly happen. (Pat says there's a probability of 1 that the sun will rise tomorrow. Being a stickler, Paul says that the probability of sunrise is very close to 1. Just because it has risen a trillion times in a row doesn't absolutely guarantee it will happen again.)

A probability of 1/2 says that you are likely to get heads half the time that you flip a coin. People also sometimes say that the probability of getting heads is fifty-fifty. The fraction, 1/2, can be converted into a decimal number. Divide 2 into 1 and you get .5. Convert this to a percentage by multiplying by 100. You get 50%. Heads will turn up 50% of the time, which is why people say the chances are fifty-fifty.

The probability of getting heads on any coin toss is 1/2—even if you've tossed the coin ten times and it has come up heads every time. You still have a fifty-fifty chance of getting heads on the eleventh toss. Each coin toss is independent of all the others.

In their heart of hearts, most people don't believe this. In "Pretty Maggie Moneyeyes," the casino owner (a man you would expect to understand probability quite well) says, "But no one can win thirty-eight thousand dollars on nineteen straight jackpots off one slot machine; It's an impossibility."

That's not quite so. Such a winning streak may be unlikely, but it's not impossible.

When Paul is speaking about probability to a large group, he has everyone stand up, take out a coin, and start flipping it. If a person's coin comes up tails, that person sits down. If it comes up heads, the person keeps flipping. Half of the audience sits down after one flip, half of the remaining half sits down after two flips. If he started with 500 people, he stops them after six flips. At that point, he has about eight people still standing. Each of them is holding a coin that has come up heads six times in a row.

What will happen when these people flip their coins one more time? Will they all get tails because those coins are "due" for a tail? Will they all get heads because they are "on a roll"? The answer, of course, is somewhere between the two extremes. Most likely four people will get heads and four will get tails. But that isn't guaranteed. After all, this is about probability, not certainty.

A flip of a coin has two possible outcomes: heads or tails. A roll of a pair of dice opens up a bigger world of possibilities.

A single die is a cube with six sides and a number from one to six on each side. Assuming the dice aren't loaded, each number has an equal chance of coming up. The probability of getting any number is one out of six or 1/6.

Now suppose you roll two dice. With two dice, you can roll numbers from two to twelve.

Here's something to think about. With one die, you were just as likely to roll a 2 as you were to roll a 5. Now that you are rolling two dice, are you just as likely to roll so that the sum of the die is 2 as you are to roll so that the sum is 5?

Take a look at the chart (on the left), and you'll see all the possible outcomes when you roll two dice. Count up how many ways that you could roll a 2. Now count up how many ways you could roll a 5.

There's only one way to roll 2. Snake eyes—a 1 on Die A and a 1 on Die B. But there are four ways to roll 5. You could roll 1 on Die A and 4 on Die B; you could roll 2 on Die A and 3 on Die B; you could roll 3 on Die A and 2 on Die B; or you could roll 4 on Die A and 1 on Die B.

Looking at the chart, you can figure out the probability of rolling a particular number. If you count up the number of possible outcomes shown on the chart, you'll see that there are 36 possible outcomes.

Now count up how many ways you can roll a particular number. There are, for instance, six ways to roll a 7. The probability of rolling a 7 is 6 out of 36 or 6/36. That's the same as 1/6. So the probability that you'll roll a 7 is 1 out of 6.

Good information to know if you want to play craps in Vegas. But does it help you with the Monty Hall Problem?

So what's the probability that you've chosen the right door? When Monty offers you a choice of three doors, the probability that you'd pick the right one is 1 out of 3 or 1/3. There are three doors and only one of them is right.

After Monty opens one door—one of the doors that does not hide the prize you want—you have a choice of two doors. If you wanted to count up the probability at this point, you might say: there are two doors and the prize is equally likely to be behind one as it is to be behind the other. So you might say that the probability that you've got the right door is 1 out of 2 or 1/2.

Unfortunately that just confirms the adage that a little knowledge is a dangerous thing and that counting is not as easy as it looks. Actually, at this point, the probability that you have the right door is still 1/3. But the probability that the prize is behind Door Number Two, the door you didn't choose, is 2/3.

How can that be? Well, that's exactly what Pat said when this problem was described to her at an Exploratorium brainstorming meeting. She didn't believe it, so she went home, sat down with her husband and three pieces of paper labeled 1, 2, and 3, and worked through all the possibilities.

First, suppose you decide not to switch. The table below shows you the possibilities:

So if you don't switch, you win one time out of three. Compare that to what happens if you always switch.

If you chose not to switch, you win one times out of three. If you chose to switch, you win two times out of three. Experimentally, it's clear that probability favors switching. Of course, experimentation is not proof for a mathematician. But Pat and Paul aren't mathematicians. We won't attempt a rigorous mathematical proof, but we will take our discussion a little farther.

In an effort to help understand how opening a door could shift the odds, Pearl Tesler, a science writer at the Exploratorium, proposed this scenario. Suppose Monty starts out with one hundred doors and he asks you to choose one. After you make your choice, he opens a door, shows you that the trip to Hawaii isn't behind that one, and offers to let you switch. You say no, and he opens another door. No tropical vacation behind that one, either. Do you want to switch?

Monty continues like this—opening door after door after door—until he is down to two doors: the door you chose and one other door. Do you want to switch? Or do you want to bet that you made the right choice the first time? Under these circumstances, most people would switch, figuring that they couldn't have chosen the right door out of one hundred possible choices.

We started this discussion with Las Vegas and we're now coming back to the topic central to that town of neon lights and rich possibilities. Any discussion of probability leads inevitably to gambling.

But you don't have to go to Las Vegas to gamble. A little closer to home, Paul and Pat can gamble on the California lottery. There are various games in the lottery, but we decided to analyze SuperLotto Plus, in which the jackpot has ranged from $7 million to more than $50 million. To play, you pick five numbers from 1 to 47 and one MEGA number from 1 to 27 and match them to the numbers drawn by the Lottery every Wednesday and Saturday. On Saturday, August 11, 2001, when we were writing this article, the SuperLotto Plus Jackpot was 14 million dollars.

So what's the probability of plunking down a dollar and winning the 14 million? Well, you have 1 chance in 47 of choosing a correct number with the first number you pick. With your second number, you have 1 chance in 46 of choosing the right number. (You've already chosen a number, so you've eliminated one from the possibilities.) With the third number, you have 1 chance in 45. And so on through the first five numbers. Then you have 1 chance in 27 of choosing the right number for the MEGA number.

To get the probability that you'll get all the numbers right and in the right order, you multiply these independent probabilities together. So you get:

1/47 x 1/46 x 1/45 x 1/44 x 1/43 x 1/27 = 1/4,969,962,360

That's means there's about 1 chance in 5 billion that your numbers will come up and that they'll be in the order in which you chose them.

Fortunately, the order of the numbers doesn't matter. There are 5! (pronounced 5 factorial) or 5x4x3x2x1 ways to order 5 different numbers. That means there are 120 different ways to order the numbers. To get our improved odds, we multiply 1 chance in 5 billion by 120. This improves the odds to about 1 chance in 4 x 10^7 or one chance in 40 million.

The odds aren't good. But do we hear some optimists in the crowd saying "Hey, playing the game only costs a buck and if you win, you will win 14 million." So what do you think? Is it worth playing?

To calculate the value of a bet, you multiply the probability that you'll win by the value of the prize you'll win. If the jackpot is 14 million and the probability of winning is 1 in 40 million then the payoff is worth $14million/40million = 0.35 dollar or thirty-five cents. Making the bet costs you a dollar. So it is not worth playing.

When the payoff reaches $40 million, the bet is almost worthwhile. Not quite worthwhile, since you might have to split the pot with some other lucky soul!

But here's another possibility. There are about 40 million different number combinations. If the payoff goes above 40 million dollars, you could just buy tickets with all the possible combinations. That seems like a sure way of winning. If you have all the combinations, one of your combinations has to win.

But wait! There is a catch. (There is always a catch.) What if someone else shares your winning number? Then you get only half of the payoff, but you'd still be out the $40 million you spent on tickets. And keep in mind: the payoff is spread over 20 years! There is also the possibility of error. If you miss a couple of combinations and one of them wins, you'll be out 40 million dollars! And the sheer magnitude of keeping track of 40 million cardboard squares boggles the mind. Better stick to a real job. (On the other hand, if you have 40 million to risk on the lottery, you don't need a real job!)

Of course, the probabilities we calculate depend on the assumption of true randomness in the drawing of the lottery numbers, the fall of the dice at the craps table, and the calculations of the computer chip that generates random numbers in the slot machine. If supernatural forces intervene (in the form of superhuman control of the dice or the assistance of a lost soul trapped in a slot machine) then the odds we calculate are meaningless.

Though we know the odds and don't rely on the supernatural, Paul and Pat still play the games in Vegas, now and then. Paul plays poker and doesn't lose too quickly. Being a keen observer of people actually helps him win a bit. Although he does live by one rule: "If you look around the poker table and can't spot the sucker then it must be you!"

Regarding gambling as a form of entertainment, Pat plays the nickel slots and gets into conversations with chain-smoking, gray-haired ladies who explain things to her. "The slots aren't paying today," says one. "You gotta play nine lines," says another. "You play three lines and you're throwing your money away."

Pat believes she's right—as gray-haired ladies so often are. If you play the nickel slots without supernatural assistance, you are certainly throwing your money away.

To learn more about Pat Murphy's science fiction writing, visit her web site at www.brazenhussies.net/murphy. For more on Paul Doherty's work and his latest adventures, visit www.exo.net/~pauld.