Because all scenarios have P(red) >= 50%, the combined P(red) >= 50%. This holds for both SIA and SSA.
The statement above is only true if the stopping condition is that “If we get to the batch B we will not roll dice but instead only make snakes with red eyes”, or in other words B must have been selected because it resulted in red eyes.
Where B is selected as the stopping condition independent of the dice result, the fate of batch B is still a dice roll so there exists a B+1 scenario. Any scenario stopping less than B must have stopped because snakes with red eyes were made, however batch B could result either in 2B−1 snakes with red eyes with probability d=1/36 or 2B−1 snakes with blue eyes with probability (1−d)=35/36. The fact that snakes are more likely to be blue eyed in batch B than they are to be red combined with the exponential weight of batch B is not with out consequence.
Note that sequence of weights provided the table are the summation terms required to find expected number of snakes created. A sum product of the series of the number of players in each scenario { 1, 3, 7, … , 2n−1 } and the frequency of terminating at a given scenario due to death {d(1−d)i−1} for i=1→n. Note: however that the sum of the scenario frequencies is d×∑ni=1(1−d)i−1=1−(1−d)n. This sum is less than 1, so we need to recover the missing (1−d)n probability which comes from the scenario where all snakes have blue eyes, our B+1 scenario/term required to find the expected number of snakes.
The statement above is only true if the stopping condition is that “If we get to the batch B we will not roll dice but instead only make snakes with red eyes”, or in other words B must have been selected because it resulted in red eyes.
Where B is selected as the stopping condition independent of the dice result, the fate of batch B is still a dice roll so there exists a B+1 scenario. Any scenario stopping less than B must have stopped because snakes with red eyes were made, however batch B could result either in 2B−1 snakes with red eyes with probability d=1/36 or 2B−1 snakes with blue eyes with probability (1−d)=35/36. The fact that snakes are more likely to be blue eyed in batch B than they are to be red combined with the exponential weight of batch B is not with out consequence.
Note that sequence of weights provided the table are the summation terms required to find expected number of snakes created. A sum product of the series of the number of players in each scenario { 1, 3, 7, … , 2n−1 } and the frequency of terminating at a given scenario due to death {d(1−d)i−1} for i=1→n. Note: however that the sum of the scenario frequencies is d×∑ni=1(1−d)i−1=1−(1−d)n. This sum is less than 1, so we need to recover the missing (1−d)n probability which comes from the scenario where all snakes have blue eyes, our B+1 scenario/term required to find the expected number of snakes.