There are multiple ways of representing odds around the world. Before today I thought there were two, then I easily learnt that Decimal and American are different, and they googling for an image of who uses what system I came across the table below. I have grabbed descriptions of all of them from here, in the spoiler below the image.

After thinking about it for a few minutes, the list of questions/calculations/algorithms I want to use odds to answer are:

We can examine the simplicity of the calculations with some examples. We could use the current US political odds, here are some. I have had to calculate American with an online converter, and the Hong Kong myself (as Decimal - 1).

For each question above I have done an example sum and given a winner and loser in my opinion:

I have put $100 on Trump to win, if he does, I will have X money (Trump Wins | 8/13 | 1.61 | -163 | 0.61):

Harris Wins = Harris Nominated * (Dems win | Harris)

Obama Wins = Obama Nominated * (Dems win | Obama)

3.3 = 1.15 * (Dems win | Harris)

27 = 17.7 * (Dems win | Obama)

3.3/1.15 = (Dems win | Harris)

27/17.7 = (Dems win | Obama)

(Dems win | Obama) / (Dems win | Harris) = (27/17.7) / (3.35/1.15)

A way to count them is 1 for win, -1 lose, 0.5 if joint:

I think I agree with the outcome, if not the method. Decimal does very easily convert to and from probabilities. I think Fraction and Hong Kong work well for the sort of profit/lose calculations that you should be doing when gambling, with each having pluses and minus's (though I think I do prefer Hong Kong now I know about it).

American is awful. Why have two different systems depending on the odds, that are so hard to compare? I wonder if it is just to make it harder for the punters to give the bookies an advantage.

**Spoiler**

*Odds description*:

DECIMAL ODDS

Decimal (or European) odds are most popular among the bookmakers online. They’re simple to understand and easy to calculate the return. That’s what the decimal odds represent – the return of your bet. For example, if the decimal odds are 1.5 and your stake is $10, you get $15 ($5 profit + $10 stake).

FRACTIONAL ODDS

Fractional (or British) odds are popular in the United Kingdom. They represent the profit, not the return. For example, if the fractional odds are 2/1 and your stake is $10, you’d get $20 profit in case of a win (plus, $10 stake back).

MONEYLINE/AMERICAN ODDS

Moneyline (or American) odds are popular among the United States bookmakers. The Moneyline odds show how much you need to bet to win $100 (if Moneyline is negative) or how much you would earn if your stake were $100 (if Moneyline is positive). For example, if the Moneyline odds are -150, you need to bet $150 to win $100; if the odds are +150, you need to bet $100 to win $150.

HONG KONG ODDS

Hong Kong (or HK) odds are standard among the bookmakers in China. They are similar to fractional odds, except written not in a fractional but decimal form instead. For example, if the Hong Kong odds are 0.25 and your stake is $10, you’d get $2.5 profit in case of a win (plus, $10 stake back).

INDONESIAN ODDS

The Indonesian odds are popular among bookmakers available in Indonesia. They are similar to Moneyline odds, except instead of taking $100 stake/profit into account, they consider 1 unit. For example, if Indonesian odds are -1.5, you need to bet $1.50 to win $1; if the odds are +1.5, you need to bet $1 to earn $1.50.

MALAY ODDS

The Malay odds are similar to American and Indonesian odds. The positive odds show the 1 unit profit while the negative odds show how much you need to bet to win 1 unit. For example, if Malay odds are -0.25, you need to bet $0.25 to win $1; if the odds are +0.25, you need to bet $1 to earn $0.25.

IMPLIED PROBABILITY

Implied probability converts the odds to probabilities. In other words, it shows what the chances of that particular bet to win are. For example, if your bet has an implied probability of 80%, it’s more likely that the bet will result in a win.

Decimal (or European) odds are most popular among the bookmakers online. They’re simple to understand and easy to calculate the return. That’s what the decimal odds represent – the return of your bet. For example, if the decimal odds are 1.5 and your stake is $10, you get $15 ($5 profit + $10 stake).

FRACTIONAL ODDS

Fractional (or British) odds are popular in the United Kingdom. They represent the profit, not the return. For example, if the fractional odds are 2/1 and your stake is $10, you’d get $20 profit in case of a win (plus, $10 stake back).

MONEYLINE/AMERICAN ODDS

Moneyline (or American) odds are popular among the United States bookmakers. The Moneyline odds show how much you need to bet to win $100 (if Moneyline is negative) or how much you would earn if your stake were $100 (if Moneyline is positive). For example, if the Moneyline odds are -150, you need to bet $150 to win $100; if the odds are +150, you need to bet $100 to win $150.

HONG KONG ODDS

Hong Kong (or HK) odds are standard among the bookmakers in China. They are similar to fractional odds, except written not in a fractional but decimal form instead. For example, if the Hong Kong odds are 0.25 and your stake is $10, you’d get $2.5 profit in case of a win (plus, $10 stake back).

INDONESIAN ODDS

The Indonesian odds are popular among bookmakers available in Indonesia. They are similar to Moneyline odds, except instead of taking $100 stake/profit into account, they consider 1 unit. For example, if Indonesian odds are -1.5, you need to bet $1.50 to win $1; if the odds are +1.5, you need to bet $1 to earn $1.50.

MALAY ODDS

The Malay odds are similar to American and Indonesian odds. The positive odds show the 1 unit profit while the negative odds show how much you need to bet to win 1 unit. For example, if Malay odds are -0.25, you need to bet $0.25 to win $1; if the odds are +0.25, you need to bet $1 to earn $0.25.

IMPLIED PROBABILITY

Implied probability converts the odds to probabilities. In other words, it shows what the chances of that particular bet to win are. For example, if your bet has an implied probability of 80%, it’s more likely that the bet will result in a win.

After thinking about it for a few minutes, the list of questions/calculations/algorithms I want to use odds to answer are:

- How much money will I get if I win the bet, aka. the return from the odds
- "I have made the bet and have no money, if I win, I will have X money"

- How much money do I gain by making and winning the bet, aka. the profit from the odds
- "If take the bet and win, I will be X better off than if I did not take the bet"
- In business speak I think this would be the "return on investment" on a successful project
- In game theory terms this would be one value in your outcome matrix (the other being the value of the bet as a loss)

- What probability of winning would make it so the bet is "fair", aka. implied probability from the odds
- "I need to think the probability of an event happening is X for it to be worth taking the bet"

- What are the "fair" odds, aka. odds from the implied probability
- "I think the true probability of an event is Y, I should take odds over X"
- "We are betting with friends, and the probability of the outcome is Y, we should make the odds X"

- Given two odds, what can we say about their relative likelihood?
- "How much less likely is Obama than Clinton to get the nomination?"

- Given multiple odds, what can we say about the odds of the difference between them?
- "What do the odds of Harris/Obama for the nomination/presidency say about their expected chances?"

We can examine the simplicity of the calculations with some examples. We could use the current US political odds, here are some. I have had to calculate American with an online converter, and the Hong Kong myself (as Decimal - 1).

Bet | Fraction | Decimal | American | Hong Kong |

Trump Wins | 8/13 | 1.61 | -163 | 0.61 |

Harris Wins | 23/10 | 3.3 | 230 | 2.3 |

Obama Wins | 26 | 27 | 2600 | 26 |

Republicans Win | 1/2 | 1.5 | -200 | 0.5 |

Harris Nominated | 2/13 | 1.15 | -667 | 0.15 |

Obama Nominated | 167/10 | 17.7 | 1670 | 16.7 |

Clinton Nominated | 40 | 41 | 4000 | 40 |

For each question above I have done an example sum and given a winner and loser in my opinion:

**Spoiler**

*Some example maths worked out*:

I have put $100 on Trump to win, if he does, I will have X money (Trump Wins | 8/13 | 1.61 | -163 | 0.61):

- 8/13 * 100 + 100 = $161
- 1.61 * 100 = $161 ## WINNER
- 1 - (100 / (-163)) = $161.34 ## LOSER
- 0.61 * 100 + 100 = $161

- 23/10 * 100 = $230
- (3.3 - 1) * 100 = $230 ## LOSER
- 230 = $230 ## WINNER
- 2.3 * 100 = $230

- One in 26 ## Joint WINNER
- One in (27 - 1) ## Joint LOSER
- One in (2600 / 100) ## Joint LOSER
- One in 26 ## Joint WINNER

- 1/(1+1/2) = 0.66
- 1/1.5 = 0.66 ## WINNER
- -(-200)/(-(-200-100)) = 0.66 ## LOSER
- 1/(1+0.5) = 0.66

- 1/0.25 = 4 ## Joint WINNER
- 1/0.25 + 1 = 5 ## Joint LOSER
- 1/0.25 * 100 = 400 ## Joint LOSER
- 1/0.25 = 4 ## Joint WINNER

- (40+1)/(167/10+1) = 2.316384
- (41)/(17.7) = 2.316384 ## WINNER
- 1670/4000 = 2.316384 ## LOSER
- (16.7+1)/(40+1) = 2.316384

- ((26+1)/(16.7+1)) / ((2.3+1)/(0.15+1)) = 0.5315871
- (27/17.7) / (3.3/1.15) = 0.5315871 ## WINNER
- I am not even trying ## LOSER
- ((26+1)/(16.7+1)) / ((2.3+1)/(0.15+1)) = 0.5315871

**Spoiler**

*Relevant bit of the table and working for decimal*:

Bet | Fraction | Decimal | American | Hong Kong |

Harris Wins | 23/10 | 3.3 | 230 | 2.3 |

Obama Wins | 26 | 27 | 2600 | 26 |

Harris Nominated | 2/13 | 1.15 | -667 | 0.15 |

Obama Nominated | 167/10 | 17.7 | 1670 | 16.7 |

Harris Wins = Harris Nominated * (Dems win | Harris)

Obama Wins = Obama Nominated * (Dems win | Obama)

3.3 = 1.15 * (Dems win | Harris)

27 = 17.7 * (Dems win | Obama)

3.3/1.15 = (Dems win | Harris)

27/17.7 = (Dems win | Obama)

(Dems win | Obama) / (Dems win | Harris) = (27/17.7) / (3.35/1.15)

A way to count them is 1 for win, -1 lose, 0.5 if joint:

Fraction | 0.5 |

Decimal | 2 |

American | -3.5 |

Hong Kong | 0.5 |

I think I agree with the outcome, if not the method. Decimal does very easily convert to and from probabilities. I think Fraction and Hong Kong work well for the sort of profit/lose calculations that you should be doing when gambling, with each having pluses and minus's (though I think I do prefer Hong Kong now I know about it).

American is awful. Why have two different systems depending on the odds, that are so hard to compare? I wonder if it is just to make it harder for the punters to give the bookies an advantage.

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