# Range of Outcomes vs. Projections: How To Evaluate Market Prices

## Why utilizing distributions is better than projections when evaluating markets.

By | December 08, 2021

Pop quiz: the closing line for Zach Lavine’s assists at Fanduel for the 12/2 Bulls-Knicks game was over/under 3.5 with -110 on both sides. What was his projected average assists for the game according to the closing line?

Your snap answer is probably 3.5, since the payout was the same on both sides of the bet. If the payout is even on both sides, the average outcome has to be the same as the over/under number, correct? Wrong- and understanding why is critical for evaluating how favorable any bet is, whether it’s spreads, totals, props, or even yes/no propositions. We’ll walk through this example to illustrate a concept that may sound scary at first- *range of outcomes*- but is something that you’ve probably internalized already, and you may not even know it.

If you’ve played any kind of fantasy sports (DFS, regular fantasy football, etc.), you might have heard phrases referring indirectly to range of outcomes: this player has a high floor/low ceiling, this is a boom-or-bust player, etc. Range of outcomes refers to the idea that a player’s projected performance in any category isn’t just described by their average outcome, it’s described by all the chances of every outcome. A completely hypothetical case: let’s say wide receiver A has a 50% chance of 0 receptions and a 50% chance of 10 receptions (an exaggerated version of a boom-or-bust player), and wide receiver B has a 50% chance of 4 receptions and a 50% chance of 6 receptions. Here’s what the math says for their average number of projected receptions:

Receiver A: 50%*0 + 50%*10 = 5

Receiver B: 50%*4 + 50%*6 = 5

Both players have the same average projected receptions, even though their range of outcomes are very different. Now, let’s say someone was offering a prop market on their receptions, where their over/under line was 4 with -110 on each side. Both receivers have an average projection of 5, and 5 is higher than the line of 4, so we should bet the over on both sides, correct? Here’s how the expected value of that math shakes out:

Over on receiver A: 50% win (10 receptions) * (10/11 payout) - 50% lose (0 receptions) *(-1 payout) = -4.5% EV

Over on receiver B: 50% win (6 receptions) * (10/11 payout) - 50% push (4 receptions) * (0 payout) = +45% EV

Admittedly, no player’s range of outcomes are as binary as this example, but it does illustrate that the relationship between average outcomes and range of outcomes is not as straightforward as the math may appear- especially when it comes to betting on said range of outcomes.

Back to the above example with Lavine’s assists: what exactly is implied by the over/under line of 3.5 assists with even payouts on both sides? Strictly mathematically, it says that Lavine’s chances of scoring exactly 0, 1, 2, or 3 assists is just as likely as Lavine scoring 4 or more assists. This implies there is a *distribution*, a fancy mathematical term for the full range of outcomes, that satisfies this condition. Here’s one such distribution, based on historical averages on players who had similar assist props:

The probabilities for each outcome less than 4 assists sum to 50%, and the probabilities for 4 or more assists sum to 50% as well. Now, what is the average outcome of this distribution? We can get that by summing the expected value of each of the outcomes (0*3.1% + 1*10.8% + 2*17.3%+ etc.) I’ll spare you the full math, but the average outcome for this distribution is 3.74 assists, a little higher than the prop market over/under value of 3.5.

Why does all of this matter in any practical sense? If you were comparing average projections to prop markets, using average projections could steer you into making mistakes. If you projected Lavine’s assists to be 3.74 and you saw his over/under was 3.5, you might be tempted into thinking that since the projected 3.74 is higher than the over/under of 3.5, the over would be a good bet. But as we just outlined, over/under 3.5 assists is a fairly priced line for an average projection of 3.74, meaning neither side of the bet would be a good one. This hammers home an important point for evaluating all bets, but most especially the prop markets: **you need to think in terms of distributions and range of outcomes, not just average outcomes**.

Does thinking in terms of distributions mean you should have a projected distribution instead of a projected average for every market you’re evaluating? Ideally, yes- and I know how exhausting that sounds. Unless you have your own simulator or other tool to spin up distributions, this is largely impractical to do on your own. Fortunately, the sports betting tools market is heating up with solutions that will let you do exactly this. Unabated’s prop tool comes from some of the sharpest minds in the betting tools space; they’re an excellent place to start. And needless to say, the tools we’re building at Betscope will be anchored by distributions and range of outcomes to ensure you’re getting the best line evaluation process possible. In the next article, we’ll walk through an example of how incorporating distribution-based thinking can help you spot value in the sports betting markets.