On Sunday, the Kansas City Chiefs lost to the Tennessee Titans 35-32. Three different times, the Chiefs kicked field goals on fourth down with three or fewer yards to go.

I’m going to do my best to explain why that was arguably the worst thing the Chiefs did in what ended up being a *terrible, horrible, no good, very bad day*.

Imagine you have a coin.

Now imagine that every time you flip the coin, two things can happen: if the coin lands *heads*, a magic genie appears and hands you $100. But every time the coin lands *tails*, the same genie shows up and takes $90 back.

Knowing that these are your two outcomes — and that the coin has an equal chance to land heads or tails — would you be willing to flip this coin?

Well, you should be! Why?

Because on average, *each two coin flips net you $10*. You flip the coin once, and it’s heads. *Yay!* You just won $100. You flip the coin again, and it’s tails. *Bummer. *The genie takes back $90. But *that still leaves you with $10 more than you had* before you started. As you keep flipping the coin, you’ll gain $10 for every two flips.

**Genie Coin Toss**

Flip | Result | Payout | Total |

1 | Heads | 100 | 100 |

2 | Tails | -90 | 10 |

3 | Heads | 100 | 110 |

4 | Tails | -90 | 20 |

Over four flips, you’d gain $20. In other words, you can *expect* to win $5 for every single flip.

Smart people have figured out this can be distilled to a formula that calculates the *expected value* of the coin flip. We multiply the *probability* of each outcome by the *value gained* — and then add those numbers together:

- 0.50 (probability of heads) X $100 (value gained for heads) = $50
- 0.50 (probability of tails) X -$90 (value gained for tails) = -$45

When you add the two results ($50 and -$45) the *expected value* of the genie’s coin flip is $5.

We know, of course, that coin flips don’t always alternate between heads and tails. It’s not that hard to flip a coin a bunch of times have have it come up tails five times in a row. So you’d probably want to have a few hundred bucks in your pocket to start!

But we also know that if you flip a coin 100 times, it’s very likely you’d get heads and tails *very close* to 50 times each. And if you’ve found a genie who is willing to make this deal — congratulations! Make 100 flips and you’ll be about $500 ahead.

But instead of one coin, let’s say we have two — a red coin and a blue coin.

If the red coin is heads, the genie hands over $490. If it’s tails, the genie gives you $20. But there’s a catch: this isn’t a normal coin. The red coin has a 55%** **chance to land on heads and only a 45% chance for tails.

But if the blue coin lands on heads, the genie gives you $220 — and if it’s tails, the genie takes $48 out of your pocket. This blue coin is even crazier, too. 90%** **of the time it lands on heads — and just 10% of the time on tails.

So if you can only flip one coin — either the red or the blue — which would you rather flip?

Yeah... I don’t know, either. But you’ve already learned how to figure it out.

Here’s the red coin:

- 0.55 (probability of heads) X $490 (value gained for heads) = $269.50
- 0.45 (probability of tails) X $20 (value gained for tails) = $9

So the *expected value* for a single flip of the red coin is $278.50 ($269.50 + $9).

Here’s the blue coin:

- 0.90 (probability of heads) X $220 (value gained for heads) = $198
- 0.10 (probability of tails) X -$48 (value gained for tails) = -$4.8

For a single flip, the blue coin’s *expected value* is $193.20 ($198 - $4.8).

So go ahead: flip the red coin!

But on Sunday — when faced with a fourth-and-2 from on his opponent’s 12-yard line — Chiefs head coach Andy Reid chose to flip the blue coin.

It may be hard to see how a coin flip compares to a football game. But if you can imagine a world where a genie hands you money (or takes it away) when a you flip a coin that doesn’t land on heads or tails an equal percentage of the time, you can see how it’s possible to calculate the best course of action when a complex variety of outcomes can take place.

Many factors influence the outcome of any given football play; the odds and values of those outcomes are less straightforward than in our simplified examples. But the logic used to make the* *decision that best improves your team’s chance to win the game* *is not remarkably different than our examples.

To decide the best of course of action for Reid at fourth-and-2 at the Titans 12-yard line, here’s what we’d need to know:

- The
*probability*of the fourth-down play succeeding - The
*value gained*if the play succeeds - The
*value gained*if the play fails - The
*probability*of the field goal succeeding - The
*value gained*if the field goal succeeds - The
*value gained*if the field goal fails

There’s no genie on the football field to tell us these values. But if the team has an analytics department — *or in this case, some nerd on the Internet* — we can make some pretty good guesses.

The *probability* part is pretty straightforward, Since 2009, 55% fourth-and-2 attempts have been converted. Similarly, about 90% of field goals have been made from that distance.

But what about the *value gained *from* *each option Reid was facing?

For that, we turn to our handy friend *expected points*. Expected points actually uses the same framework we’re discussing here — *expected value — *to determine how much value a team should expect to get when they’re at any position on the field.

If you convert the first down, you’ll be on the opponent’s 10-yard line, which is worth about 4.9 points. That makes sense, right? There’s a good chance you’ll score a touchdown; it’s just not guaranteed.

But if you fail on a fourth-and-2 at the 12 — handing the ball back to the opponent — that’s worth *negative* 0.2 points to them, because they will have to drive all the way down the field to score — and the most likely outcome of any drive is a punt. So for the team holding the ball on the fourth-and-2, the *failure* of the conversion *adds* 0.2 points.

What about making (or missing) the field goal?

You’d think it would just be three points when it succeeds — but it’s not quite that simple. After the field goal is made, the scoring team kicks off. Most of the time, the opposing team gets the ball around the 25-yard line. That is worth 0.8 points for the *other* team, so the value of the field goal is three points *minus* 0.8 — or 2.2 points.

But if the field goal fails, the other team gets the ball back at the spot of the kick — which in this case would be around their own 20. That’s worth 0.48 points to them — and therefore, *minus* 0.48 points to the team that missed the field goal.

Any of these numbers sound familiar to you?

That’s right. Going for it on fourth-and-2 at the 12 is the red coin. Kicking the field goal is the blue coin. I just multiplied the expected points numbers by 100 to get you excited about winning $500 from a genie. As Andy Reid would say... that’s how I roll.

Flipping the *red* coin would have given the Chiefs a better chance to win. But Reid flipped the *blue* coin... because that’s how *he* rolls.

This all assumes the Chiefs have just an average offense — and the Titans have an average defense. It assumes that weather does not effect the chance of making the field goal. It doesn’t take the ability of Harrison Butker into account. It also takes for granted that only the minimum amount of yards needed for a first down will be gained, leaving the Chiefs at the 10-yard line instead of even closer — or in the end zone.

But do we really think that taking all those particulars into account would make the decision for a field goal *more *likely? Probably not. The Chiefs have one of the best offenses in football and had a fourth-down conversion rate in the top five last season — not to mention the top ten this season.

No matter which way you slice it, kicking this field goal left points on the field. And it turns out that the numbers work out similarly for all three of the Chiefs’ field goals before the fourth quarter.

A lot of crazy things happened on Sunday: a Patrick Mahomes jump-pass that will likely end up in his Hall of Fame highlight reel, a botched long snap that gave the Titans a chance at victory, a wide-open dropped pass by a star receiver... you name it!

But to me, none of those are crazier than a billion-dollar organization failing at a very simple task: giving their head coach the information he needs to make the best decision in a fourth-down situation — over and over and over again.

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