ࡱ> =?<q`,bjbjqPqP.f::$DDDDDDDX    X      ". :jllllll$h DB  BBDD  BD D jBjDD  Џ\ Xd0nDhBBBBBBBtBBBBBBBXXX XXX XXXDDDDDD  Calculators for computing bridge game probabilities Calculators provide data for arriving at probabilistically optimum decisions on whether it is advisable, in a given situation, to bid a contract to make, pass, double, redouble, or sacrifice (both in IMPs or MPs competitions). There are seven Calculators with different functions. 1. Calculator CALC1E The calculator is designed to help players to take probabilistically optimal decisions in IMPs competitions. It allows, within certain limits, to calculate a threshold probability value of making contract (P1). It is assumed that Team I, playing the first contract, would make it or go down just one trick. Team II in other room would bid the same contract, as Team I or 1 level lower. Calculation of the threshold probability value P1 is based on an ideal model, when in both rooms high-level players make no mistakes (like playing double-dummy). It is also assumed that contract, bid by Team I, can be made with probability P or can be defeated by 1 trick with probability 1 P (IMP scale is being used). To calculate the threshold probability value P1 the user must select and enter for both Teams the denomination, level and vulnerability of the contract and whether it is doubled or redoubled. When all entries are completed, the calculator 1 generates the threshold probability value of making contract (P1). The symbol < before the threshold probability value P1 means that Team I will win the board if the probability P of making the contract is less than P1; the symbol > before the calculated value P1 means that Team I will win the board if the probability P exceeds P1. If, instead of a number, the calculator shows the symbol L, it means that Team I will always lose/tie the board. The table presents values of P at which Team I will win (Win) or lose/tie (Lost) the given board. To come to the optimum decision the player should compare calculated threshold probability value P1 to the probability P which he/she estimates on the basis of his/her own experience, taking into account all features of this given board. Calculators CALC2E and CALC2A The calculators CALC2 and CALC2A are designed to help players to take probabilistically optimal decisions to bid a defensive or aggressive contract to save (to sacrifice) or to make (both in MPs or IMPs competitions). The results for each board are being calculated on the assumption that high-level players make no mistakes (like playing double-dummy). To calculate the possible results a given board the user must select and enter for both Pairs the denomination, level, vulnerability and possible result of the contract and whether it was doubled or redoubled. When all entries are completed, the calculator generates the table, in which the vertical axis shows the probabilities of making Contract 1. If the probability is 0, Pair 1 will go down 1 (1st case) or make an overtrick (case 2). If the probability is 1, Pair 1 will always make the contract. On the horizontal axis fractions are plotted (from left to right), where the numerator represents the probability of Contract 2 made, going down by 1, 2 or 3 tricks, and the denominator represents the probability of Contract 2 going down by 1, 2, 3 or 4 tricks accordingly. The table presents MPs, which Pair I will win (Win) or lose/tie (Lost) for a given board at different probabilities of making contracts. To come to the optimum decision the player should estimate real probabilities of making contracts on the basis of his/her own experience, taking into account all features of this given board. The calculator provides data for arriving at probabilistically optimum decisions at IMPs. In that case this given board will be won (Win) or lost/tied (Lost) by the Team, whose opponents decided to bid a contract to save (to sacrifice). QUANTITATIVE ESTIMATION OF BIDDING DEFENSIVE CONTRACTS FOR IMPs OR MPs COMPETITIONS 1) IMPs competitions. The first pair bids a first contract and can make it with probability p1 or lose it with probability 1 - p1. In first case the first pair will win A1IMP's and in second case will lose A2 IMP's (where A2 <0). Average IMP's value for the first pair is  EMBED Equation.3  (1) The second pair in the same deal bids the another (higher or defensive) contract and can make it with probability p2 or lose it with probability 1 - p2. In first case second pair will win B1 IMP's and in second case will lose B2 IMP's (where B2 <0). Average IMP's value for the second pair is  EMBED Equation.3  (2) It is obvious, that if the second pair will win S2 IMP's the first pair will lose S2 IMP's. The first pair will win the given deal only if  EMBED Equation.3  or if  EMBED Equation.3  (3) 2) MPs competitions. The first team bids on the first table first contract and can make it with probability p1 or lose it with probability 1 - p1. In first case first team will win A1 MP's and in second case will lose A2 MP's. The first team bids on the second table second (higher) contract and can make it with probability p2 or lose it with probability 1 - p2. In first case first team will win B1 MP's and in second case will lose B2 MP's. If both contracts on both tables are statistically independent events average total quantity MP's, win by the first team in the given deal, is  EMBED Equation.3  (4) Or  EMBED Equation.3  (5) It is obvious, that the first team will win the given deal only when  EMBED Equation.3  Comparing (3) and (5), we will see, that formulas for IMPs and MPs competitions are completely coincide. Calculator CALC3E The calculator is designed to calculate possible results on a given board for Team 1, when Team 1 bids game and Team 2 bids a small or grand slam, depending upon denomination, vulnerability and level of contracts bid by Team 1 and Team 2 and whether it is doubled or redoubled and probabilities pi (p1 probability of just making the contract bid by Team 1, p2 - probability of making the contract bid by Team 1 with just 1 overtrick, p3 - probability making the contract bid by Team 1 with just 2 overtricks, p4 - probability of making the contract bid by Team 1 with just 3 overtricks, where p1 + p2 + p3 + p4 = 1). Calculator CALC4E The calculator is designed to calculate possible results on a given board for Pair 1 depending upon different probabilities P of making the contract bid by Pair 1 and selected probabilities Pi of making the contract bid by Pair 1 (P1 - probability of the contract bid by Pair 2 making, P2 - probability of the contract bid by Pair 2 going down by 1, P3 - probability of the contract bid by Pair 2 going down by 2, P4 - probability of the contract bid by Pair 2 going down by 3, where p1 + p2 + p3 + p4 = 1). Calculator CALC5E The calculator is designed to calculate the probability P1 of Kings presence in E-W-hands (number of Kings), the probability P2 of Aces presence in E-W-hands (number of Aces) and the probability P3 of Kings and Aces presence in E-W-hands (number of Aces and Kings) for selected number of honors cards in N-hand and HP of S-hand. When all entries are completed after clicking CALCULATE the calculator provides the probability values P1 and/or P2 and/or P3. Calculator CALC6E The calculator is designed to calculate the shape probability and includes three different calculators: Calculator 1, Calculator 2 and Calculator 3. The calculator 1 is designed to calculate (for selected sum number of cards in all suits of N and S hands): a) the probability of E(W)-shape, b) most probable E(W)-shape, c) the relative probability (the ratio of the probability of selected E(W)-shape to the probability of the most probable E(W)-shape, d) number of possible E(W)-shape, and e) the probability of suit distribution. To calculate a), b), c) and d) the user must select the sum number of cards in all suits of N and S hands and E(W)-shape and click CALCULATE. To calculate the probability of suit distribution the user must select the sum number of cards in all suits of N and S hands and number of cards in any E(W)-suit and click CALCULATE. The calculator 2 is designed to calculate the most probable shapes of three other hands and the probability of W/E void or singleton in any suit for any given N-shape and completely or partially selected S, W,E-shape and to calculate the probability completely or partially selected S-shape for any completely selected N-shape. To calculate the user must completely select N-shape and completely or partially select S-shape (1 or 2) and/or W-shape and/or E-shape and click CALCULATE. The calculator 3 is designed to calculate the probability S-shape for any completely selected N-shape and selected maximum and minimum number of cards for different suits of S-hand. To calculate the user must completely select N-shape, maximum, minimum and exact number of cards for one, two, three or all suits of S-hand and click CALCULATE. To open worksheets click Macro Security. In Macro Settings choose Enable all macros.  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