Hi everyone and welcome to my new column, Dear Plus EV! I’ll be answering questions I get from the community on the analytics and mathematics of betting. Do you have a question that you would like to have answered? DM me on Twitter @PlusEVAnalytics, or email me at [email protected]. Some questions may be lightly edited for clarity. Let’s get right to it!
When originating player props, do you find either mean or median to be a more useful measure in terms of forecasting? – Twitter @nickconnell_
This is one of the most commonly misunderstood concepts in the quantitative betting world. When using statistics to bet, our minds tend to focus on the mean because it’s easily accessible, but any bet that’s constructed as an over/under should really be analyzed using the median, not the mean.
You’ll find lots of situations in sports statistics where the median and the mean differ significantly from each other, usually with the median lower than the mean. For example, in Derrick Henry’s last full season (2020) he averaged 127 rushing yards per game. That average was pushed upward by THREE (!) 200+ yard rushing games. That same season, his median was 114 rushing yards per game. Against a league-average defense and with a rushing prop of over/under 120 yards, using the mean would lead you to wrongly conclude that the over is the better side, while the median would correctly steer you to the under.
Based on my understanding, beating the closing line consistently should yield a profit in sports betting. So would it be logical to always bet with a book that has slightly off market numbers assuming the vig is identical? For example, using the Betstamp app, BUF is -2.5 (-105) across most sportsbooks however my book has BUF -1.5 (-105). If I bet all of these instances, over what approximate sample size would I likely turn a profit? - Twitter @inTEJrity
Let’s tackle this question in two parts.
Part 1: What is the EV of a -1.5 -105 in the NFL if the market is at -2.5 -105/-115?
There are tools to do this all over the place, my personal favorite is Unabated. If I go into their alternate line calculator and put in these values, click “remove vig,” and then “get alternate line price," I see the vig-free price for -1.5 is -105. So in this particular case, your -1.5 -105 has an EV of zero. Bet it if you want some free entertainment, but that’s all the value you’ll get as you will break even over the long run. The answer to “should I always bet off-market numbers” is “only if the number is off-market by enough to overcome the vig.”
This makes the second part of the question largely irrelevant, so I’m going to take some creative license here. What if our book was offering -1.5 -102 instead of -1.5 -105? Then you could calculate the market implied win probability as 105/205 = 51.2% and the EV as 51.2% x (202/102) – 1 = +1.4%. Now we can continue…
Part 2: How many repeated same-sized bets with 51.2% win probability paying -102 would it take to likely turn a profit?
At the risk of stating the obvious, it depends on your definition of “likely.” Make this bet only once, and you’re 51.2% likely to turn a profit. Make it over and over a huge number of times, and the law of large numbers tell us that you will approach (but never exactly achieve) 100% certainty of positive profit. Neither of those two answers is very satisfying, so let’s deal with the space in between.
Let N = the number of bets.
The number of wins has a binomial distribution with parameters N and 0.512. Let’s call that W.
Then, your net profit after N bets is W*(202/102) – N. So, your net profit is positive if W > N * 102/202.
We can use Excel’s binomial distribution function to calculate the probability of your net profit being positive after N bets:
=1-BINOM.DIST(N*102/202,N,0.512,TRUE)
Let’s calculate this for a few different values of N:
N = 1 → 0.512
N = 5 → 0.522
N = 10 → 0.407
N = 100 → 0.556
N = 1,000 → 0.682
N = 10,000 → 0.921
N = 100,000 → 0.999996
So what have we learned here? The law of large numbers requires some pretty large numbers before it truly kicks in. This type of advantageous - but barely - bet is very typical of what you might regularly come across if you’re chasing stale or off-market lines, and it might be eye-opening to see that after a thousand bets you’re still going to be in the red almost one third of the time!
P.S. to quote Sesame Street, one of these things is not like the other…what’s the deal with N = 10 at 0.407? It’s because with N = 10, there’s a relatively high probability that the outcome is 5 wins and 5 losses, which would put you ever so slightly on the wrong side of zero profit. To extend this point, try it yourself with N = 2!
I was just curious what your thoughts were on, respectively, the Fangraphs MLB model and the 538 MLB model (and NFL model for that matter). Do you think their numbers are good? – Twitter @jakeell48874660
There are many examples of published models whose predictions are way off-market. 538 in particular is notorious for this. I’m sure all of these models have very smart people behind them (well, maybe except for 538). But, when the market and someone's model are giving me different numbers, I don’t care if that guy is Einstein…unless I can see the inner workings of the model and understand why it would beat the market, give me the market number 100 times out of 100.
Why do I feel this way? Because nobody is perfect and no model is perfect. The difference is that errors in market numbers are self-correcting because people are constantly betting into them and moving them. This creates a “wisdom of the crowd” mechanism where the collective market is smarter than any individual participant. No such self-correction exists in any of these other numbers, so any errors they make persist forever.
To summarize, I would not recommend using any of these published models in your betting. To the extent that they ever had any predictive value at all, that value would be already priced in by the time it gets to you.
I’m trying to learn +EV betting. I know this is +EV but how do I calculate the actual percentage? (attached: a screenshot of a Pokerstars odds boost for Bo Bichette to record a run & TOR Blue Jays to win at +900) - Twitter @thecadpromokid
Same game parlays are notoriously difficult to price. Not only do you need to know the true odds of Bichette scoring a run and of the Jays winning, but you need to quantify the correlation between these two events. They’re more likely to happen together – Bichette scoring a run increases the Jays changes of winning – which makes your parlay more likely to win than what you’d calculate by treating them as independent events.
There are two ways we can approach this problem. The hard way is by building a model that evaluates the probability of these events happening together. Unless you plan on betting these regularly for large amounts, that’s probably not worth the effort. Fortunately it seems like you’ve already discovered the easy way, because your DM to me included a screen shot from FanDuel showing this exact same game parlay with odds of +160. In order to offer the kind of customizable same game parlay products that many of these books have nowadays, the books would have needed to have these complicated models already built.
Is the +160 correct just because FanDuel says it is? Of course not, but unless you want to do this the hard way it’s one of those places where you might want to settle for “close enough”. The one thing we do need to do is remove the vig from that +160 number…which is also no easy task because we don’t know how much vig is in these lines nor do we know whether it’s distributed evenly or shaded. We know that same game parlays have a crap-ton of vig in them, and that they likely overcharge you for correlated outcomes. We also know that the vig on any parlay increases with the number of legs, so it would be less for 2 legs than for a larger parlay. Based on that, I’m going to make a wild guess of 15% theoretical hold on FanDuel’s +160. This means that the actual payout of 2.60 (ALWAYS use decimal odds when doing this calculation!) is 85% of what a fair payout would be, so 2.60 / 0.85 = 3.06 is the fair payout and 1 / 3.06 = 32.7% is the implied win probability.
Finally, you can calculate the EV on the Pokerstars bet as 32.7% * 10.00 – 1 = +227%. These boosted odds promos usually have very low limits, but fire away for as much as you’re allowed!