1. BETTING RATES - CAN THEY BE USEFUL? Magdalena Hykšová CTU Prague 2011

2. Which pair has the greates chance for the win in the competition StarDance 2010? e.g., Kříž/Stodůlková: stake 1 EUR  win 2,55 EUR or nothing

3. Similarly in the following cases:

4. Where should we bet for the football match Spain – Honduras at the world championship 2010? Betting shop1 (S wins)X (draw)2 (H wins) Bet-at-home1,118,0015,00 Fortuna1,126,7013,00 Tipsport1,138,1015,05 Numbers in the table: how much the bookmaker’s pays for 1 EUR in the case of the sake on the right result

5. Where should we bet for the football match Spain – Honduras at the world championship 2010? Betting shop1 (S wins)X (draw)2 (H wins) Bet-at-home1,118,0015,00 Fortuna1,126,7013,00 Tipsport1,138,1015,05 Whatever happens, Tipsport pays the most The highest chance for the win: Spain

6. We bet for all alternatives to obtain 100 EUR in any case. How many percent of the stake will we get back? Betting shop1X2Return Bet-at-home1,118,0015,00 Fortuna1,126,7013,00 Tipsport1,138,1015,05 Return – e.g., for Fortuna: To win 100 EUR in the case of the win of Spain, we have to bet 100/1,12 on it In total we pay:

7. We bet for all alternatives to obtain 100 EUR in any case. How many percent of the stake will we get back? Betting shop1X2Return Bet-at-home1,118,0015,0091,53 % Fortuna1,126,7013,0089,36 % Tipsport1,138,1015,0593,04 % Return – e.g., for Fortuna: In total we pay:

8. betting rate p... how much the bettor has to bet on the event A to win 1 EUR, if A occurs Hockey championschip 2010, final CZE – RU p CZE = 1/4, p RU = 9/10 To win S = 100 CZK, I have to bet 100  1/4 = 25 EUR... win of the Czech Republic 100  9/10 = 90 EUR... win of Russia

9. betting rate p... how much the bettor has to bet on the event A to win 1 EUR, if A occurs Hockey championschip 2010, final CZE – RU p CZE = 1/4, p RU = 9/10 To win S = 100 CZK, I have to bet 100  1/4 = 25 EUR... win of the Czech Republic 100  9/10 = 90 EUR... win of Russia I will pay 115 EUR, win 100 EUR  return:

10. Hockey championschip 2010, final CZE – RU p CZE = 1/4, p RU = 9/10 To win S = 100 CZK, I have to bet 100  1/4 = 25 EUR... win of the Czech Republic 100  9/10 = 90 EUR... win of Russia I will pay 115 EUR, win 100 EUR  return: p CZE + p RU > 1  the bettor loses  fair bett

11. FAIR BETT the bookmaker specifying p must be in danger of getting into the role of the better  we admit positive and negative betts values the bettor pays: p CZE  S + p RU  S, obtains: S p CZE + p RU > 1... bettor chooses S < 0 and wins p CZE + p RU 0 and wins  p CZE + p RU = 1

12. FAIR BETT S... the value of the win if A occurs (positive or negative) Bettor’s profit: Z(A) = S – pS = (1 – p) S Z(  A) = – pS In total we can write: Z = (|A| – p) S For n events: Z = (|A 1 | – p 1 ) S + (|A 2 | – p 2 ) S +... + (|A n | – p n ) S

13. Bookmaker’s estimation of probability P(A): e.g., P(A) = 0,5 no danger of changing the roles  it is profitable to propose e.g. p = 0,75 Consider S = 100 win 100 CZK if A occurs... bett 75 EUR repetition: bookmaker’s guess about bettor’s profit ca. 1/2 cases: A occurs  Z = 100 – 75 = 25 EUR ca. 1/2 cases: A does not  Z = – 75 EUR Average profit:

14. Bookmaker’s estimation of probability P(A): Mean bettor’s profit: S > 0  mean bettor’s profit Z < 0  bookmaker benefits  fair bett Fair bett: we allow S < 0 p... probability that the bookmaker frankly assigns to the event A

15. BETTING RATES DETERMINATION bettor’s profit: Z(A) = S – pS = (1 – p) S Z(  A) = – pS coherent betting rates: profit Z cannot be always positive, no matter which event occurs (sure win of the bettor) coherence requirement  basic axioms of probability theory i.e., to prevent a sure loss, bookmaker’s rates p have to satisfy axioms of probability theory

22. BETTING RATES  Warning – who can profit from them?  Application of probability – determination of rates (although the bets are not fair)  The meaning of axioms of probability theory  Broadening of probability conception

23. SUBJECTIVE INTERPRETATION PROBABILITY = measure of personal conviction or belief in the occurence of some event Václav Šimerka (1818 – 1887), 1882 Frank Plumpton Ramsey (1903 – 1930), 1931 (1926) Bruno de Finetti (1906 – 1985), 1931, 1937 Leonard Jimmie Savage (1917 – 1971), 1954

24. SUBJECTIVE INTERPRETATION Everyday considerations No traffic jam probably occurs on the highway this time. Our hockey team has no chance against Russia. This drug should be effective in your case. The election will be probably won by CSSD. If I set out at 16:10, I will perhaps catch the train. CNB predicts the 1.8% VAT grow in 2012.