Odds to Probability Question: bookmaker puts odds of event occuring at 6/4

In a recent article, I read:

"The U.K. bookmaker Paddy Power has the odds of Trump being impeached before the end of his term at 6/4..."

So to get the probability that Trump will be impeached, I used the relationship

P = odds/(1+odds), so P = (6/4)/(1+6/4)) = 0.60 or 60%

But the sentence in the article continues:

"...or what the U.K. and Ireland betting houses see as a 40% chance."

What am I missing here?

tstone said:

But the sentence in the article continues:

"...or what the U.K. and Ireland betting houses see as a 40% chance."

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Before I can really respond to what you are asking you need to state what the 40% chance represents. Is it the chance that Trump does not get impeached, the chance that Trump does get impeached or the chance that it will rain tomorrow?

Well, if you piece together the whole unedited sentence from the article, it reads:

"The U.K. bookmaker Paddy Power has the odds of Trump being impeached before the end of his term at 6/4, or what the U.K. and Ireland betting houses see as a 40% chance."

I don't see how to read that except that they see a 40% chance that Trump will be impeached.

tstone said:

Well, if you piece together the whole unedited sentence from the article, it reads:

"The U.K. bookmaker Paddy Power has the odds of Trump being impeached before the end of his term at 6/4, or what the U.K. and Ireland betting houses see as a 40% chance."

I don't see how to read that except that they see a 40% chance that Trump will be impeached.

Click to expand...

You are correct to be troubled as I too see that it should have said 60% instead of 40%.

tstone said:

Well, if you piece together the whole unedited sentence from the article, it reads:

"The U.K. bookmaker Paddy Power has the odds of Trump being impeached before the end of his term at 6/4, or what the U.K. and Ireland betting houses see as a 40% chance."

I don't see how to read that except that they see a 40% chance that Trump will be impeached.

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My guess would be that when they said "the odds of ..." they meant "the odds against ...". The latter is a very common way to state odds, and very many people misunderstand the terminology so that they would not see the difference. And 6:4 odds against impeachment means a 40% probability of impeachment.

OK, I guess it comes down to a matter of semantic interpretation. Could definitely confuse some bettors, though, IMO.

Thank you both for your insights. I

tstone said:

OK, I guess it comes down to a matter of semantic interpretation. Could definitely confuse some bettors, though, IMO.

Thank you both for your insights. I

Click to expand...

I found the article (by searching for your quote) and found that they later say (referring to another source), "Among bettors using BetDSI, Trump is now an overwhelming favorite to be impeached before 2021. The odds are 1/5 or -500. That means a bettor would have to risk $500 just to win $100 on a bet that Trump will be impeached." Clearly this is referring to what I would call odds of 1:5 against. I'd never heard of expressing odds as "-500", but I find both forms explained here: https://www.gamblingsites.org/sports-betting/beginners-guide/odds/ . Clearly betting uses different terminology than mathematicians. Some quotes:

"Moneyline odds are also known as American odds, and this is the format most commonly used in the United States. They can be displayed as either a positive or a negative number. A positive number expresses how much a correct wager of $100 would win, while a negative number expresses how much you would need to stake in order to win $100. If you saw odds of +150, you would know that a $100 bet could return $150 in winnings, plus the initial stake of $100. If you saw -150, you would know you need to stake $150 to return $100 in winnings, plus the initial stake of $150. An even money wager (where you stand to win an amount equal to your stake) is expressed as +100."

"Decimal odds used to be associated mostly with mainland Europe, Canada, and Australia. However, they have now largely become the standard at most online bookmakers with the exception of some US betting sites. This is because they are the most straightforward of the three formats and are expressed simply as a single positive number, typically to two decimal places. The number shows how much the total payout will be, including the original stake per unit staked. For example, a winning bet at 1.5 would return a total of $1.50 for every $1 staked. A winning bet at 2.25 would return a total of $2.25 for every $1 staked. An even money bet is expressed as 2.00."

"Fractional odds ... are displayed as a fraction. A simple example is 3/1, which is said as "three to one". 5/1 is said as "five to one", and so on. With 3/1, you can win three units for every one unit staked, and with 5/1 you can win five units for every one unit staked. 1/1 is even money, so you can win one unit for every unit staked." They use the word "against" only in some circumstances, but the order of the numbers as written is always against:for, as I would express it.

Maybe some of you who deal with actual betting will have more to say about this language.