# Implied Probability of an Outcome -BetFair Edition

&#8220-Does prediction market guru [= Chris Masse] understand probabilities?&#8220-, asks our good friend Niall O&#8217-Connor.

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Let&#8217-s ask economics PhD Michael Giberson:

Yes, I think you are right. I just looked at your exchange with Niall and Niall&#8217-s post, and haven&#8217-t thought through just how the over-round may affect things.

But it seems okay to do it just the way you say, because the digital odds implies a precise numerical prediction and that prediction can be stated in the form of a probability. Call the calculated number an implied probability of the event, and then you don&#8217-t have to worry that a complete group of related market prices don&#8217-t add to 100 percent.

If a trader believes that event X should be trading at 70 percent and sees current digital odds of 1.56 at Betfair ( =&gt- 64.1 percent), he should buy (considering fees, etc.). If the digital odds move to 1.4 ( =&gt- 71.4 ) then sell or at least don&#8217-t buy.

Niall may be hung up on using a pure concept of probability. The purity is not useful- your explanation is useful. You win.

(Feel free to quote from this email, should you wish.)

-Mike

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UPDATE: Michael Giberson precises his comment&#8230-

Niall, I agree that Professor Sauer&#8217-s presentation explains how to estimate true probabilities from odds that do not sum to one. I was taking Chris Masse to be explaining a related, but slightly different task: the conversion of the digital odds that Betfair quotes to an implied probability.

The point of my slightly snide comment concerning purity reflects the pragmatic view that a trader could use the method Chris describes to convert from digital odds to an implied probability (which may be easier for some traders to think with and trade on). A single quote of digital odds implies a particular probability estimate. Chris&#8217-s math gets the trader from the one number to the other. (=useful to traders)

To get to the estimate of true probabilities, as you have explained, a trader must have a complete set of odds for all possible outcomes for an event. This additional information requirement would completely stymie a trader wishing to arrive at the true probability estimates in cases in which some of the data is unavailable. (= not as useful to traders)

Read the previous blog posts by Chris. F. Masse:

• Pervez Musharraf prediction markets –Eric Zitzewitz Edition
• The Over-Round Explained
• WHY THE PREDICTION MARKETS WILL LIKELY F**K UP SUPER TUESDAY 2008.
• Still unconvinced by prediction market journalist Justin Wolfers
• Oprah Winfrey
• RIGHT-CLICK THIS IMAGE, AND FILL IN THIS SURVEY, PLEASE.
• Papers on Prediction Markets

# BetFair Digital Odds = BetFair Probabilities

Odds that Hillary Clinton gets the 2008 Democratic nomination = 1.56 (digital odds taken at 9:15 AM EST)

To get the implied probability expressed in percentage:

• Take the number &#8220-1&#8243–
• Divided it by the digital odds (here &#8220-1.56&#8243-)-
• Then multiply the result by 100-
• 64.1% = ( 1 / 1.56 ) x 100

BetFair-generated implied probability is not far away from InTrade&#8217-s 62.1%.

Psstt&#8230- This present post was prompted by Niall O&#8217-Connor, who puts all his faith in the BetFair instant &#8220-over-round&#8221- &#8212-which indeed doesn&#8217-t add up to the virgin and perfect &#8220-100%&#8221- that Niall is seeking (like the Monthy Python were seeking the Holy Grail). Good luck for your quest, Niall.