Rule of 15Conservative pass hand (hands with 11 HCP or less and 12 HCL points or less) Since this is a rule for whether you should open the bidding at fourth seat, the result of opening these hands should be better then the average pass out hands and better then the other side. The average number of tricks for hands just pass the rule of 15 is 8.8 vs. 8.2. The first test is really not a fair test. Everyone will open the hand with 15 HCP and void in spade. This type of hand will make the result bias towards the rule of 15. Let us restrict ourselves to hands that satisfy the rule of 15 and have 11 or less high card length points. The average number of tricks for those hands is 8.478 with the other side taking 8.043 tricks. It has positive score on 62.7% Let us look at the worse case of 11 HCP and 4333 with 4 spades. The average number of tricks for those hands is 7.93 with the other side taking 8.103tricks. Wait a minute; the rule of 15 still wins in this case. Even though it makes less tricks but it has positive score 51.4%of the time. The reason is very simple. The rule of 15 produces a spade trump suit 33% of the time vs. the opponent having the spade trump suit only 10% of the time. When you have the spade trump suit you have a positive score if you can take the same number of tricks. Let us look at the worse case of 10 HCP and 5332 with 5 spades. The average number of tricks for those hands is 8 with the other side taking 8.3 tricks. The rule of 15 did not work in this case. It has positive score 49.4%of the time. Let us look at the worse case of 9 HCP and 6322 with 6 spades. The average number of tricks for those hands is 8.2 with the other side taking 8.7 tricks. The rule of 15 did not work in this case. It has positive score 46%of the time. Let us look at the second worse case of 9 HCP and 6331 with 6 spades. The average number of tricks for those hands is 8.43 with the other side taking 8.74 tricks. The rule of 15 did not work in this case. It has positive score 48.5%of the time. Let us look at the worse case of 8HCP and 7222 with 7 spades. The average number of tricks for those hands is 8.64 with the other side taking 9.1 tricks. The rule of 15 did not work in this case. It has positive score 47.5%of the time. Aggressive pass hand (hands with 11 HCP or less and 11 HCL points or less) This small change makes a big difference in some cases. The reason is very simple. There are 3 pass out hands and two of those are opponents. The better the pass out hands can be the less often you should open in 4th seat. Since this is a rule for whether you should open the bidding at fourth seat, the result of opening these hands should be better then the average pass out hands and better then the other side. Let us restrict ourselves to hands that satisfy the rule of 15 and have 11 or less high card length points. The average number of tricks for those hands is 8.3 with the other side taking 7.8 tricks. It has positive score on 65% Let us look at the worse case of 11 HCP and 4333 with 4 spades. The average number of tricks for those hands is 7.72 with the other side taking 7.84 tricks. It has positive score 52.1%of the time. Let us look at the worse case of 10 HCP and 5332 with 5 spades. The average number of tricks for those hands is 8 with the other side taking 8.3 tricks. The rule of 15 work in aggressive pass hand condition and did not work in the conservative pass hand condition. It has positive score 52.6%of the time. Let us look at the worse case of 9 HCP and 6322 with 6 spades. The average number of tricks for those hands is 7.97 with the other side taking 8.25 tricks. The rule of 15 work in aggressive pass hand condition and did not work in the conservative pass hand condition. It has positive score 51.1%of the time. Hands with 12 HCP fails the rule of 15 I always open hands with 12 or more HCP and did not consider looking at those hands for the rule of 15 until Paul Hightower mention it in his post. Let us look at two worse case of 12 HCP hands that fails the rule of 15. For 12 HCP, 2 spade and no 5 card suits The average number of tricks for those hands is 8.45 with the other side taking 7.53 tricks. It has positive score 67.6% of the time for aggressive pass condition. The average number of tricks for those hands is 8.56 with the other side taking 7.52 tricks. It has positive score 65% of the time for conservative pass condition. For 12 HCP, 1 spade and no 5 card suits (1444) The average number of tricks for those hands is 8.98 with the other side taking 7.44 tricks. It has positive score 73% of the time for aggressive pass condition. The average number of tricks for those hands is 9.13 with the other side taking 7.62 tricks. It has positive score 63% of the time for conservative pass condition. Should open all 12 HCP hands except very bad 12 HCP that is down graded to 11 HCP
