Meanwhile, the ever-oblivious Keltham is turning to gesture at his Prestidigitated whiteboard:

All-10s (2 YES, 3 NO):

Propensity:     Likelihood:
---------------     --------------
1/10                (1/10)^2 * (9/10)^3 =   729/10000
2/10                (2/10)^2 * (8/10)^3 = 1024/10000
4/10                (4/10)^2 * (6/10)^3 = 3456/10000
5/10                                               = 3125/10000
6/10                (6/10)^2 * (4/10)^3 = 2304/10000
8/10                                               =   512/10000
9/10                                               =     81/10000

All-14s (3 YES, 2 NO):

                                                      = same but table flipped

"Right, so, in the first row, we've got the hypothesis, what if all-10s have a 0.1 'propensity' towards YES."

"Where, in this context, what that means, is:"

"Suppose the world is such a way that a random person from our 'survey-pool', about whom we know only that they have 10s in all mental stats of Intelligence, Wisdom, and Charisma, has a 10% 'propensity' to guess the correct answer on the 2-4-6 challenge within 30 minutes.  Where 'propensity' is sort of a metaphysically fraught concept.  But for our purposes in the kid's version of the lesson, 'suppose 10% YES propensity in all-10s' means, suppose the world itself is such that, from our perspective, it works out to 10% of the all-10s guessing within 30 minutes.  Nethys, maybe, would know that some people in that pool were 99% likely to get it and some were 0.1% likely to get it, so it's not that the 10% chance of YES is an objective property of every single person.  But from our perspective, the world is such a place that Nethys would say that around 10% of the people in that pool were 99% likely to get it, and we don't know which people."

"That's one hypothesis, the one in the first row.  The second row's hypothesis is that all-10s in our survey-pool have a 20% propensity towards YES when we'll test them using our current experimental setup."

"Suppose we test five all-10 subjects - one after another, rather than in a simultaneous session - and see these exact results in order:  NO, YES, YES, NO, NO."

"After getting a NO result with the first all-10 subject, we've seen something that was 90% likely in the world where all-10s have a 10% propensity to YES.  But 80% likely in the world where all-10s have a 20% propensity to produce YES results."

"After seeing the YES from the second subject, we've now seen an additional fact that was 10% likely in the world where all-10s have a 10% YES propensity, and 20% likely in the world with 20% propensity."

"The two facts combined, that we've seen so far, are then 0.9 * 0.1 = 0.09 likely in the 0.1-propensity world, and 0.8*0.2 = 0.16 likely in the 0.2-propensity world."

Meanwhile, the ever-oblivious Keltham is turning to gesture at his Prestidigitated whiteboard:

All-10s (2 YES, 3 NO):

Propensity:     Likelihood:
---------------     --------------
1/10                (1/10)^2 * (9/10)^3 =   729/100000
2/10                (2/10)^2 * (8/10)^3 = 1024/100000
4/10                (4/10)^2 * (6/10)^3 = 3456/100000
5/10                                               = 3125/100000
6/10                (6/10)^2 * (4/10)^3 = 2304/100000
8/10                                               =   512/100000
9/10                                               =     81/100000

All-14s (3 YES, 2 NO):

                                                      = same but table flipped

"Right, so, in the first row, we've got the hypothesis, what if all-10s have a 0.1 'propensity' towards YES."

"Where, in this context, what that means, is:"

"Suppose the world is such a way that a random person from our 'survey-pool', about whom we know only that they have 10s in all mental stats of Intelligence, Wisdom, and Charisma, has a 10% 'propensity' to guess the correct answer on the 2-4-6 challenge within 30 minutes.  Where 'propensity' is sort of a metaphysically fraught concept.  But for our purposes in the kid's version of the lesson, 'suppose 10% YES propensity in all-10s' means, suppose the world itself is such that, from our perspective, it works out to 10% of the all-10s guessing within 30 minutes.  Nethys, maybe, would know that some people in that pool were 99% likely to get it and some were 0.1% likely to get it, so it's not that the 10% chance of YES is an objective property of every single person.  But from our perspective, the world is such a place that Nethys would say that around 10% of the people in that pool were 99% likely to get it, and we don't know which people."

"That's one hypothesis, the one in the first row.  The second row's hypothesis is that all-10s in our survey-pool have a 20% propensity towards YES when we'll test them using our current experimental setup."

"Suppose we test five all-10 subjects - one after another, rather than in a simultaneous session - and see these exact results in order:  NO, YES, YES, NO, NO."

"After getting a NO result with the first all-10 subject, we've seen something that was 90% likely in the world where all-10s have a 10% propensity to YES.  But 80% likely in the world where all-10s have a 20% propensity to produce YES results."

"After seeing the YES from the second subject, we've now seen an additional fact that was 10% likely in the world where all-10s have a 10% YES propensity, and 20% likely in the world with 20% propensity."

"The two facts combined, that we've seen so far, are then 0.9 * 0.1 = 0.09 likely in the 0.1-propensity world, and 0.8*0.2 = 0.16 likely in the 0.2-propensity world."

Meanwhile, the ever-oblivious Keltham is turning to gesture at his Prestidigitated whiteboard:

All-10s (2 YES, 3 NO): Propensity:     Likelihood: ---------------     -------------- 1/10                (1/10)^2 * (9/10)^3 =   729/100000 2/10                (2/10)^2 * (8/10)^3 = 1024/100000 4/10                (4/10)^2 * (6/10)^3 = 3456/100000 5/10                                               = 3125/100000 6/10                (6/10)^2 * (4/10)^3 = 2304/100000 8/10                                               =   512/100000 9/10                                               =     81/100000 All-14s (3 YES, 2 NO):                                                       = same but table flipped

"Right, so, in the first row, we've got the hypothesis, what if all-10s have a 0.1 'propensity' towards YES."

"Where, in this context, what that means, is:"

"Suppose the world is such a way that a random person from our 'survey-pool', about whom we know only that they have 10s in all mental stats of Intelligence, Wisdom, and Charisma, has a 10% 'propensity' to guess the correct answer on the 2-4-6 challenge within 30 minutes.  Where 'propensity' is sort of a metaphysically fraught concept.  But for our purposes in the kid's version of the lesson, 'suppose 10% YES propensity in all-10s' means, suppose the world itself is such that, from our perspective, it works out to 10% of the all-10s guessing within 30 minutes.  Nethys, maybe, would know that some people in that pool were 99% likely to get it and some were 0.1% likely to get it, so it's not that the 10% chance of YES is an objective property of every single person.  But from our perspective, the world is such a place that Nethys would say that around 10% of the people in that pool were 99% likely to get it, and we don't know which people."

"That's one hypothesis, the one in the first row.  The second row's hypothesis is that all-10s in our survey-pool have a 20% propensity towards YES when we'll test them using our current experimental setup."

"Suppose we test five all-10 subjects - one after another, rather than in a simultaneous session - and see these exact results in order:  NO, YES, YES, NO, NO."

"After getting a NO result with the first all-10 subject, we've seen something that was 90% likely in the world where all-10s have a 10% propensity to YES.  But 80% likely in the world where all-10s have a 20% propensity to produce YES results."

"After seeing the YES from the second subject, we've now seen an additional fact that was 10% likely in the world where all-10s have a 10% YES propensity, and 20% likely in the world with 20% propensity."

"The two facts combined, that we've seen so far, are then 0.9 * 0.1 = 0.09 likely in the 0.1-propensity world, and 0.8*0.2 = 0.16 likely in the 0.2-propensity world."