(1) The current (that is 13.July.2017) online betting odds – and so implied probability – of Donald Trump being replaced as President of the USA before his first 4 year term ends seem to be rather high. Below are are odds taken from betting websites and calculations showing the probabilities (net of implied bookmaker’s margins) implied by those odds. *Don’t take this probabilities too seriously, and in particular if a range of probabilities is given that is merely the range given by these odds and the calculations made: the actual probabilities might be lower or higher that the naive might think implied by that range (or by a single figure).*

That said, I don’t know if Trump holds the record for the 20th and 21st century USA President with the highest first presidential year odds of being replaced before the end of his first 4 year term, but he must be a very strong contender for that record.

(1.1) Odds (and implied bookmaker’s margins) can differ quite widely. So using only Coral, which seems to have smallish spread of bookmaker’s margin of 9% to 11%, and has the advantage that odds are given for Trump being replaced in 2020 (all other odds I found were for Trump being replaced in 2020 or later), here are some odds and calculations of implied probabilities net of bookmaker’s margins as at the afternoon of 14.July.2017:

q= aproximate probability net of an estimate of bookmaker’s margin

o= *approximate* odds net of an estimate of bookmaker’s margin

O= odds including bookmaker’s margin

* Trump replaced within 12 months: q= 0.23 o= 3.5/1

* Trump replaced within 15 months (mid-term): q= 0.26 o= 3/1

* Trump replaced in 2017-2020: q= 0.45 o= 1.25/1

check: Trump To Leave Office Via Impeachment Or Resignation Before End 1st Term:

O= 11/10; q= 10/(11 plus 10) / 1.11 (margin) = 0.43

which is reasonably close to 0.45, and anyway should be smaller:

the 0.45 should be including the possibility that Trump dies in office by 2020;

* Trump replaced after 2020: q= 0.55 o= 0.8/1;

check: Trump to serve full first term: O= 8/11;

q= 11/(8 plus 11) / 1.11 (margin) = 0.52

which is reasonably close to 0.55

* May replaced within 12 months: q= 0.4 o= 1.5/1

* Next General Election within 12 months: q= 0.27 o= 2.75/1

* Trump & May replaced within 12 months: q= 0.23*0.41 = 0.094 o= 9.5/1

* Trump replaced & GE within 12 months: q= 0.23*0.27 = 0.062 o= 15/1

http://sports.coral.co.uk/politics/international/us-elections/donald-trump-specials-7055847.html

When Will Trump Be Replaced?

2017 = 5/1; Q = 1/(5 plus 1) = 0.17; q = Q/1.11 = 0.15;

2018 = 5/1; Q = 1/(5 plus 1) = 0.17; q = Q/1.11 = 0.15;

2019 = 8/1; Q = 1/(8 plus 1) = 0.11; q = Q/1.11 = 0.10;

2020 = 16/1; Q = 1/(16 plus 1) = 0.06; q = Q/1.11 = 0.05;

2021 = 11/8; Q = 8/(11 plus 8) = 0.42; q = Q/1.11 = 0.38;

2022 = 33/1; Q = 1/(33 plus 1) = 0.03; q = Q/1.11 = 0.03;

2023 = 40/1; Q = 1/(40 plus 1) = 0.02; q = Q/1.11 = 0.02;

2024 = 50/1; Q = 1/(50 plus 1) = 0.02; q = Q/1.11 = 0.02;

2025 or later = 8/1; Q = 1/(8 plus 1) = 0.11; q = Q/1.11 = 0.10;

total Q = 1.11 implying bookmaker’s margin = 11%; total q = 1.00;

Trump being replaced within 12 months q = 0.23;

approximate calculation q2017-Jun.2018 = q2017 plus 6/12 * q2018

http://sports.coral.co.uk/politics/uk/uk-politics/theresa-may-specials-6129146.html

Year May Is Replaced As Prime Minister

2017 = 5/2; Q = 2/(5 plus 2) = 0.29; q = Q/1.09 = 0.27;

2018 = 5/2; Q = 2/(5 plus 2) = 0.29; q = Q/1.09 = 0.27;

2019 = 9/4; Q = 4/(9 plus 4) = 0.31; q = Q/1.09 = 0.28;

2020 or later = 4/1; Q = 1/(4 plus 1) = 0.20; q = Q/1.09 = 0.18;

total Q = 1.09 implying bookmaker’s margin = 9%; total q = 1.00;

May being replaced as Prime Minister within 12 months q = 0.41;

approximate calculation q2017-Jun.2018 = q2017 plus 6/12 * q2018

http://sports.coral.co.uk/politics/uk/uk-politics/next-general-election-7329939.html

Year of Next General Election

2017 = 5/1; Q = 1/(5 plus 1) = 0.17; q = Q/1.11 = 0.15;

2018 = 11/4; Q = 4/(11 plus 4) = 0.27; q = Q/1.11 = 0.24;

2019 = 5/2; Q = 2/(5 plus 2) = 0.29; q = Q/1.11 = 0.26;

2020 = 12/1; Q = 1/(12 plus 1) = 0.08; q = Q/1.11 = 0.07;

2021 = 12/1; Q = 1/(12 plus 1) = 0.08; q = Q/1.11 = 0.07;

2022 or later = 7/2; Q = 2/(7 plus 2) = 0.22; q = Q/1.11 = 0.20;

total Q = 1.11 implying bookmaker’s margin = 11%; total q = 0.99;

(I’m ignoring the 0.01 rounding error in q)

Next General Election in or before Jun.2018 q = 0.27;

approximate calculation q2017-Jun.2018 = q2017 plus 6/12 * q2018

* What follows gives an indication (no more than that) of the spread of the odds amongst bookmakers.*

(2) Example calculated implied net of bookmaker’s margins probabilities for Trump ceasing to be President:

- 23% to 31% for being replaced in 12 months July.2017 to June.2018
- 26% to 36% for being replaced in 15 months July.2017 to mid-term elections November.2018
- 40% to 52% for being replaced in 2 years 6 months July.2017 to December.2019
- 45% (Coral) for being replaced before January.2021
- 55% (Coral) for surviving to January.2021

(3) Odds for Trump being replaced from Coral and Oddschecker on 13.July.2017:

Trump replacement year | Coral | BetVictor | PaddyPower | Betfair | SkyBet |

2017: | 5/1 | 4 | 3 | 34/5 | 6 |

2018: | 5/1 | 3 | 4 | 10/3 | 10/3 |

2019: | 8/1 | 11/2 | 7 | 24/5 | 4 |

2020: | 16/1 | ||||

2021: | 11/8 | ||||

2022: | 33/1 | ||||

2023: | 40/1 | ||||

2024: | 50/1 | ||||

2020 or later: | 5/6 | 4/5 | 9/10 | ||

2025 or later: | 8/1 |

(4) Calculate implied probabilities before adjusting for bookmaker’s margins

examples: 3 is 3/1 giving 1/(3+1) = 0.25; 34/5 gives 5/(34+5) = 0.13;

Trump replacement year | Coral | BetVictor | PaddyPower | Betfair | SkyBet |

2017: | 0.17 | 0.20 | 0.25 | 0.13 | 0.14 |

2018: | 0.17 | 0.25 | 0.20 | 0.23 | 0.23 |

2019: | 0.11 | 0.15 | 0.13 | 0.17 | 0.20 |

2020: | 0.06 | ||||

2021: | 0.42 | ||||

2022: | 0.03 | ||||

2023: | 0.02 | ||||

2024: | 0.02 | ||||

2020 or later: | 0.55 | 0.56 | 0.53 | ||

2025 or later: | 0.11 | ||||

==== | ==== | ==== | ==== | ||

total | 1.11 | 1.15 | 1.14 | 1.06 | |

implied bookmaker’s margin | 11% | 15% | 14% | 6% | assume 10% |

(5) True probabilities for exclusive and exhaustive events should add up to 1. But bookmaker’s need to cover their expenses and make a profit, so they must reduce their odds, which implies increasing the apparent probabilities, so the sum of the implied probabilities for exclusive and exhaustive events will add up to more than 1. See Wikipedia – Odds – Gambling odds versus probabilities and Wikipedia – Mathematics of bookmaking – Overround on multiple bets.

(6) Calculate implied probabilities net of implied bookmaker’s margin

example: Betfair 0.13/(1+6%) = 0.13/1.06 = 0.12

Trump replacement year | Coral | BetVictor | PaddyPower | Betfair | SkyBet |

2017: | 0.15 | 0.17 | 0.22 | 0.12 | 0.13 |

2018: | 0.15 | 0.22 | 0.18 | 0.22 | 0.21 |

2019: | 0.10 | 0.13 | 0.11 | 0.16 | 0.18 |

2020: | 0.05 | ||||

2021: | 0.38 | ||||

2022: | 0.03 | ||||

2023: | 0.02 | ||||

2024: | 0.02 | ||||

2020 or later: | 0.48 | 0.49 | 0.50 | 0.48** | |

2025 or later: | 0.10* | ||||

==== | ==== | ==== | ==== | ==== | |

total: | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |

* 2020 or later = 0.60 = 0.05 + 0.38 + 0.03 + 0.02 + 0.02 + 0.10

** 0.48 = 1.00 – (0.13 + 0.21 + 0.18)

(7) Other implied probabilities net of implied bookmaker’s margin

Trump replacement year | Coral | BetVictor | PaddyPower | Betfair | SkyBet |

(7.1) 2017 annualised: | 0.28 | 0.31 | 0.39 | 0.23 | 0.24 |

(7.2) Jul.2017-Jun.2018: | 0.23 | 0.29 | 0.31 | 0.24 | 0.24 |

(7.3) Jul.2017-Nov.2018: | 0.26 | 0.34 | 0.36 | 0.29 | 0.29 |

(7.4) 2017-2019: | 0.40 | 0.52 | 0.51 | 0.50 | 0.52 |

(7.5) 2017-2020: | 0.45 | ||||

(7.6) 2021 or later: | 0.55 |

- (7.1) the 2017 probability is for Trump leaving in the 6 months July-December 2017;

this annualised figure is for comparison with the 2018 and 2019 figures for full years;

example 2017 annualised for Betfair: 1.00 – (1.00 – 0.12) * (1.00 – 0.12) = 0.23 - (7.2) probability for Trump leaving in the 12 months July.2017-June.2018;

example for Betfair: 0.12 + (1.00 – (1.00 – 0.22)**0.5) = 0.24 - (7.3) probability for Trump leaving in the 15 months July.2017-November.2018;

example for Betfair: 0.12 + (1.00 – (1.00 – 0.22)**0.75) = 0.29 - (7.4) sum of probabilities for Trump leaving in 2017, 2018, 2019;
- (7.5) sum of probabilities for Trump leaving in 2017, 2018, 2019, 2020;
- (7.6) sum of probabilities for Trump leaving in 2021, 2022, 2023, 2024, 2025 or later;