逆轉勝機率之計算及其在台灣運動彩券投注策略上的應用

Abstract

本研究旨在探討逆轉勝問題及其擴展應用，依據美國職業籃球(NBA)比賽歷史資料顯示，於比賽前三節結束時落後之球隊，最終仍有機會贏得勝利。Gill(2000)假定了主、客隊的前三節得分和它們在第四節的分數皆具有常態分配，而且假設主、客隊在前三節落後之情況下，第四節得以逆轉勝的分數之條件分布為常態時，推導出逆轉勝機率之計算公式。\n然而依常理，前三節得分分差的大小對逆轉勝機率應有不同程度的影響，為更貼近真實賽情，本研究擴展了Gill(2000)模型，引入二元常態模型探討逆轉勝機率之計算公式。同時利用NBA歷年賽事資料對二元常態模型與Gill(2000)模型進行預測逆轉勝機率之比較，我們發覺二元常態模型在預測逆轉勝機率較Gill(2000)模型更精準。\n最後本研究將上述結果進一步應用至台灣運動彩券分析上，利用二元常態模型計算出投注主隊或客隊獲利金額之期望值，以作為台灣運動彩券讓分盤是否投注之決策依據。然因資料可能不是非常符合二元常態模型，以致實際平均獲利在精準度上未達理想，為保守起見，因此下注1元時的獲利金額之期望值至少0.5元時，始建議下注。在前述決策方式下，我們發現下注客隊時，可獲得平均0.1079元的利潤。

This study aims to explore the problem of turning defeat into victory and its extended application. According to the historical data of National Basketball Association, the team that falls behind on the first three quarters of the game will still have a chance to win. Gill(2000) assumes that the home team and away team scores at the end of the third quarter and during the fourth quarter are normally distributed. To give the formulas of the conditional probabilities of turning defeat into victory, Gill(2000) further assumes that each of these two corresponding conditional distributions follows a normal distribution.\nIn practice, the difference of scores during the fourth quarter depends not only on which teams falls behind but also on the difference of scores of these two teams at the end of the third quarter. Therefore, we extend Gill(2000) model to introduce a bivariate normal distribution model. We then give turning defeat into probability formulas by using this new model. Using NBA historical data, we compare our bivariate normal model with Gill(2000) model and find that the bivariate normal model is better than that of Gill(2000).\nThis study also applies our bivariate normal model to analyze Taiwan sports lottery. We calculate the expected profit of betting a team to decide whether to bet on this team through Taiwan sports lottery. Although data may not fit the bivariate normal model well and prediction is not accurate enough, we still suggest people interested in gaming could bet on the teams having the expected profit at least 0.5 NTD per 1 NTD wager. Under such circumstances, it is found that our average profit per 1 NTD betting is still 0.1079 NTD on the away team.