哈佛大学概率与数理统计课件.pptVIP

Why is Probability Theory Not Ancient? Religious beliefs Or No concept of equally likely outcomes ? ? ? Playing Fair With a Biased Coin Unequal probability of H and T: p ? ? Probability of H is p Probability of T is 1-p Toss twice and ignore pairs HH and TT Probability of HT is p(1-p) Probability of TH is (1-p)p Call combination HT ‘Newheads’ Call combination TH ‘Newtails’ Newheads and Newtails are equally likely Efficiency is poor (50%) – discard the HH and TT s Faking Random Sequences THHTHTHTHTHTHTHTHTTTHTHTHTHTHTHH THHTHTHTHHTHTHHHTTHHTHTTHHHTHTTT HTHHTHTTTHTHTHTHHTHTTTHHTHTHTHTT Winning (and Losing) Streaks The Win-Win Scenario The odds for the runners are a1 to 1, a2 to 1, a3 to 1, and so on, for any number of runners in the race. If the odds are 5 to 4 then we express that as an ai of 5/4 to 1 Bet a fraction 1/(ai +1) of the total stake money on the runner with odds of ai to 1 If there are N runners, we will always make a profit if Q = 1/(a1 +1) + 1/(a2 +1) + 1/(a3 +1) +….+ 1/(aN +1) 1 Winnings = (1/Q – 1) ? our total stake Example: Four runners and the odds for each are 6 to 1, 7 to 2, 2 to 1, and 8 to 1 and. Then we have a1 = 6, a2 = 7/2, a3 = 2 and a4 = 8 and Q = 1/7 + 2/9 + 1/3 + 1/9 = 51/63 1 Allocate our stake money with 1/7 on runner 1, 2/9 on runner 2, 1/3 on runner 3, and 1/9 on runner 4 We will win at least 12/51 of the money we staked (and of course we get our stake money back as well). The Moral The Three-Box Trick You are Twice As Likely to Win if You Switch than if You Don’t ! So having sold your diamonds what do you do with the money? Banks are far too risky, so.. The Midsummer Murders scenario Placing a bet of 100$ on outcome A with bookmaker 2 and a bet of $100 * 1.43 / 3.9 = 36.67 on outcome B at bookmaker 1 would ensure you a profit. In case outcome A comes out, one could collect 100 x 1.43 = 143 from bookmaker 2. In case outcome B comes out, one could collect 36.67 x 3.9 = 143 from bookmaker 1. One would have invested