# Pearl Jam, The Missing Dollar Riddle, and The Double-Counting Error

About a month ago, I discussed the Monty Hall Problem in an installment of The Labyrinthian. I love logic problems, so I’ve decided to do another piece dedicated to a riddle that I first heard when I was in grade school. Before that, though, let’s talk about something else that I heard first in grade school: Pearl Jam.

This is the 38th installment of The Labyrinthian, a series dedicated to exploring random fields of knowledge in order to give you unordinary theoretical, philosophical, strategic, and/or often rambling guidance on daily fantasy sports. Consult the introductory piece to the series for further explanation.

## Pearl Jam and “Dollar Short”

[INSERT here a really long and tangential introduction about Pearl Jam and its song “Dollar Short,” which turned into “Alive,” the band’s first hit.]

## The Missing Dollar Riddle

Again, I should apologize for that really long and tangential introduction about Pearl Jam and its song “Dollar Short,” which turned into “Alive,” the band’s first hit — but wasn’t it totally worth it? I mean, it’s just so interesting and necessary that I simply can’t imagine this piece without that introduction.

True, its only connection to our topic is a loose one — “Dollar Short” and “The Missing Dollar Riddle” both contain within them a hint of the same idea — but, still, I think that we can all agree that the detour was totally worth it, if for no other reason than to see someone actually put in print these words, which so many have probably thought but few have had the strength to declare publicly:

Pearl Jam is to Mother Love Bone what Foo Fighters is to Nirvana: A lesser, derivative band that ideally should not exist.

Now, I grant that it’s a douche move for me to quote myself, especially since you just read those same words in the previous section — but they’re worth repeating, no?

(By the way, I don’t really believe that PJ is worse than MLB, but I felt the pressure to say something bold in the introduction, to make you feel that reading over 5,000 words on grunge music had been worth it. Oh well.)

## The Missing Dollar Riddle (Revisited)

Let’s get to it. Versions of this puzzle or riddle, which is of a larger class called mathematical misdirection puzzles, have been around since at least the 1700s and probably a lot longer. In each iteration of the riddle, the particular subject of the mathematics changes to suit the interests of the time period, but the same logical move remains.

In 2003, this riddle somehow became something that people who didn’t know how the internet worked emailed to everyone they knew. That phenomenon was almost as frustrating as the riddle itself. In the email, after the riddle was this sentence: “Send this email to five people and the answer will appear on your screen.” Perhaps that seemed realistic to people who eight years prior had thought that Hackers was a sensible movie — but I digress.

Here’s a fairly common version of the riddle:

Three people have dinner at a restaurant. The waiter brings them the bill which is \$30, so each diner pays \$10. The waiter takes the cash to the manager at the cash register, who looks at the bill and realizes that the diners have collectively been overcharged by \$5. The manager hands \$5 to the waiter and instructs him to return the money.

On the way to the table, the waiter — knowing that \$5 cannot be split cleanly between three people — decides to return only \$3 to the diners and to keep the remaining \$2 for himself.

With \$3 having been returned, each diner has in effect paid \$9, and collectively they have paid \$27. And the waiter has kept \$2 as a tip. So that’s \$29 in total.

Where’s the missing dollar?

Send the link to this article to five people and the answer will appear on your screen.

## There is no Missing Dollar

True to my word, the answer just appeared on your screen: There is no missing dollar. The riddle is misleading.

The riddle encourages you to add together \$27 and \$2 when in reality they should not be added together. The riddle encourages you to count some numbers twice, and you definitely shouldn’t.

The answer isn’t simply, “There is no missing dollar.” The answer is also, “I reject your faulty mathematical premises.”

With the \$3 refund, the diners have indeed paid \$27, of which \$25 are in the cash register (or at least with the manager) and \$2 are with the waiter. You can see that there is no missing dollar, and that the \$2 are already accounted for within the aforementioned \$27.

In truth, this is not the Missing Dollar Riddle. It’s really just an instance of a double-counting error that the riddle passes off as sound logic.

## The Double-Counting Error

When you are experimenting in the FantasyLabs and using our Trends and Player Models tools, you need to be highly careful not to count numbers twice (or thrice). Doing so puts you at risk of relying on a trend or model that has an unrepresentative (or at least uncertain) Plus/Minus.

What do I mean by counting twice? Let’s go through some examples.

Weighted On-Base Average and Isolated Power are both valuable sabermetric baseball statistics. Although they are different — wOBA is an adjusted catch-all statistic that takes into account each type of batting event, whereas ISO a more of a measure of raw power — the truth is that the metrics have significant overlap in that they both take into account (albeit in different ways) singles, doubles, triples, and home runs. Given the importance of hits to baseball, these metrics have similarities that should not be ignored.

Likewise, if you look at our Advanced MLB Data, you will see that we have many batted-ball metrics, such as Distance, Exit Velocity, Hard-Hit Percentage, and Airtime. All of these metrics measure batting performance in different ways, yet at the same time they are all correlated. The better a batter does, the further his average batted ball will go, the faster it will travel, the harder it will be hit, and the longer it will stay in the air. Each of these metrics is useful in its own way, but the overlap they share is important.

If you create a trend or model and seeking balance, then you are at risk of committing a double-counting error if you prioritize multiple sabermetric stats or multiple advanced stats. You will be giving too much weight to metrics that measure performance in similar ways.

That’s not to say that you can’t have trends and models that are intentionally overweight on specific stats or data. If you do that for a purpose — if you want to see what happens when you leverage a set of metrics in an extreme manner, perhaps because you think you will find an exploitable edge for guaranteed prize pools — then that’s fine. That’s not the commission of an error. That’s the exploration of a tactic.

It’s only a double-counting error if you don’t think about it. Ultimately, the key is to consider what you are doing. And, before that, the key is to know what you are doing.

## Knowledge isn’t Power — But They’re Correlated

Knowledge isn’t the same as power, but the two are similar. Often those who have an abundance of one possess at least a moderate amount of the other. If you want to have DFS power and keep your DFS power, you need to know what you’re doing.

And the best way to know what you are doing, especially if you are a new subscriber and/or new to MLB DFS, is to follow these steps:

1. Cut a hole in a box.
2. Watch our MLB Video Tutorials.
3. Learn what numbers go into our metrics and experiment in the Trends and Models tools for as long as you can.

If you do not have a solid grasp of the stats and tools, you will either not use them — and you should use them! — or you will use them poorly, with the double-counting error likely to be one of the most prevalent errors that you make.

Don’t be anything less than the best version of your DFS self. Learn the numbers and the tools. Otherwise, you’ll never be more than the Pearl Jam to your inner Mother Love Bone.

———

The Labyrinthian: 2016, 38

Previous installments of The Labyrinthian can be accessed via my author page. If you have suggestions on material I should know about or even write about in a future Labyrinthian, please contact me via email, [email protected], or Twitter @MattFtheOracle.

About a month ago, I discussed the Monty Hall Problem in an installment of The Labyrinthian. I love logic problems, so I’ve decided to do another piece dedicated to a riddle that I first heard when I was in grade school. Before that, though, let’s talk about something else that I heard first in grade school: Pearl Jam.

This is the 38th installment of The Labyrinthian, a series dedicated to exploring random fields of knowledge in order to give you unordinary theoretical, philosophical, strategic, and/or often rambling guidance on daily fantasy sports. Consult the introductory piece to the series for further explanation.

## Pearl Jam and “Dollar Short”

[INSERT here a really long and tangential introduction about Pearl Jam and its song “Dollar Short,” which turned into “Alive,” the band’s first hit.]

## The Missing Dollar Riddle

Again, I should apologize for that really long and tangential introduction about Pearl Jam and its song “Dollar Short,” which turned into “Alive,” the band’s first hit — but wasn’t it totally worth it? I mean, it’s just so interesting and necessary that I simply can’t imagine this piece without that introduction.

True, its only connection to our topic is a loose one — “Dollar Short” and “The Missing Dollar Riddle” both contain within them a hint of the same idea — but, still, I think that we can all agree that the detour was totally worth it, if for no other reason than to see someone actually put in print these words, which so many have probably thought but few have had the strength to declare publicly:

Pearl Jam is to Mother Love Bone what Foo Fighters is to Nirvana: A lesser, derivative band that ideally should not exist.

Now, I grant that it’s a douche move for me to quote myself, especially since you just read those same words in the previous section — but they’re worth repeating, no?

(By the way, I don’t really believe that PJ is worse than MLB, but I felt the pressure to say something bold in the introduction, to make you feel that reading over 5,000 words on grunge music had been worth it. Oh well.)

## The Missing Dollar Riddle (Revisited)

Let’s get to it. Versions of this puzzle or riddle, which is of a larger class called mathematical misdirection puzzles, have been around since at least the 1700s and probably a lot longer. In each iteration of the riddle, the particular subject of the mathematics changes to suit the interests of the time period, but the same logical move remains.

In 2003, this riddle somehow became something that people who didn’t know how the internet worked emailed to everyone they knew. That phenomenon was almost as frustrating as the riddle itself. In the email, after the riddle was this sentence: “Send this email to five people and the answer will appear on your screen.” Perhaps that seemed realistic to people who eight years prior had thought that Hackers was a sensible movie — but I digress.

Here’s a fairly common version of the riddle:

Three people have dinner at a restaurant. The waiter brings them the bill which is \$30, so each diner pays \$10. The waiter takes the cash to the manager at the cash register, who looks at the bill and realizes that the diners have collectively been overcharged by \$5. The manager hands \$5 to the waiter and instructs him to return the money.

On the way to the table, the waiter — knowing that \$5 cannot be split cleanly between three people — decides to return only \$3 to the diners and to keep the remaining \$2 for himself.

With \$3 having been returned, each diner has in effect paid \$9, and collectively they have paid \$27. And the waiter has kept \$2 as a tip. So that’s \$29 in total.

Where’s the missing dollar?

Send the link to this article to five people and the answer will appear on your screen.

## There is no Missing Dollar

True to my word, the answer just appeared on your screen: There is no missing dollar. The riddle is misleading.

The riddle encourages you to add together \$27 and \$2 when in reality they should not be added together. The riddle encourages you to count some numbers twice, and you definitely shouldn’t.

The answer isn’t simply, “There is no missing dollar.” The answer is also, “I reject your faulty mathematical premises.”

With the \$3 refund, the diners have indeed paid \$27, of which \$25 are in the cash register (or at least with the manager) and \$2 are with the waiter. You can see that there is no missing dollar, and that the \$2 are already accounted for within the aforementioned \$27.

In truth, this is not the Missing Dollar Riddle. It’s really just an instance of a double-counting error that the riddle passes off as sound logic.

## The Double-Counting Error

When you are experimenting in the FantasyLabs and using our Trends and Player Models tools, you need to be highly careful not to count numbers twice (or thrice). Doing so puts you at risk of relying on a trend or model that has an unrepresentative (or at least uncertain) Plus/Minus.

What do I mean by counting twice? Let’s go through some examples.

Weighted On-Base Average and Isolated Power are both valuable sabermetric baseball statistics. Although they are different — wOBA is an adjusted catch-all statistic that takes into account each type of batting event, whereas ISO a more of a measure of raw power — the truth is that the metrics have significant overlap in that they both take into account (albeit in different ways) singles, doubles, triples, and home runs. Given the importance of hits to baseball, these metrics have similarities that should not be ignored.

Likewise, if you look at our Advanced MLB Data, you will see that we have many batted-ball metrics, such as Distance, Exit Velocity, Hard-Hit Percentage, and Airtime. All of these metrics measure batting performance in different ways, yet at the same time they are all correlated. The better a batter does, the further his average batted ball will go, the faster it will travel, the harder it will be hit, and the longer it will stay in the air. Each of these metrics is useful in its own way, but the overlap they share is important.

If you create a trend or model and seeking balance, then you are at risk of committing a double-counting error if you prioritize multiple sabermetric stats or multiple advanced stats. You will be giving too much weight to metrics that measure performance in similar ways.

That’s not to say that you can’t have trends and models that are intentionally overweight on specific stats or data. If you do that for a purpose — if you want to see what happens when you leverage a set of metrics in an extreme manner, perhaps because you think you will find an exploitable edge for guaranteed prize pools — then that’s fine. That’s not the commission of an error. That’s the exploration of a tactic.

It’s only a double-counting error if you don’t think about it. Ultimately, the key is to consider what you are doing. And, before that, the key is to know what you are doing.

## Knowledge isn’t Power — But They’re Correlated

Knowledge isn’t the same as power, but the two are similar. Often those who have an abundance of one possess at least a moderate amount of the other. If you want to have DFS power and keep your DFS power, you need to know what you’re doing.

And the best way to know what you are doing, especially if you are a new subscriber and/or new to MLB DFS, is to follow these steps:

1. Cut a hole in a box.
2. Watch our MLB Video Tutorials.
3. Learn what numbers go into our metrics and experiment in the Trends and Models tools for as long as you can.

If you do not have a solid grasp of the stats and tools, you will either not use them — and you should use them! — or you will use them poorly, with the double-counting error likely to be one of the most prevalent errors that you make.

Don’t be anything less than the best version of your DFS self. Learn the numbers and the tools. Otherwise, you’ll never be more than the Pearl Jam to your inner Mother Love Bone.

———

The Labyrinthian: 2016, 38

Previous installments of The Labyrinthian can be accessed via my author page. If you have suggestions on material I should know about or even write about in a future Labyrinthian, please contact me via email, [email protected], or Twitter @MattFtheOracle.

#### About the Author

Matthew Freedman is the Editor-in-Chief of FantasyLabs. The only edge he has in anything is his knowledge of '90s music.