1.A) Approximately. (Originally this was yes, until you stated that there were at least 700 million people on the planet. After that information, I updated this answer, because I realized that the problem had an additional assumption of a finite number of people, thus encountering any one left-handed person reduces the odds, very very marginally, of any different future person I encounter being left-handed, because the pool of people I’m drawing from now has slightly different odds.)
1.B) No. (Still.)
1.C) Approximately. Why the answer is different without resorting to math: In 1.B, we nonrandomly pull 10 right-handed students out of the group. In a pool of 24 10-sided die we’ve already rolled, we’ve pulled out 10 of them which did not roll 1; this does not alter the number which did roll 1, increasing their relative proportion. In this case, we’ve rolled the dice 10 times, and they never came up 1; the remaining 14 times remain fair dice rolls.
1.D) (Modified) Approximately.
1.E) Very very slightly.
2.) [Edited; apparently I screwed up when I added the possibility of an exact match] .41%, still assuming we’re not considering the proximity of 0 to 3, and including closer matches. (That is, only considering identical digit matches.)
3.ab) Supposing it’s more likely that a higher quality student is A than !A; it’s possible that it’s extremely unlikely for a person who isn’t high A to have high grades while still having more high grade students who aren’t A than are A, if the odds of A are substantially lower than the odds of being neither A nor B but still having high grades. So there’s not enough information.
Assuming it’s more likely you’re A and have high grades than ~A and have high grades, however, and assuming that this distribution holds for the grade average for each college (p(A|G) > .5 for all three G), you should in all cases favor low-B students, because the remaining pool of accepted students is more likely to be A than !A, because !B limits you to the pool of students who are either A or !A with high (enough) grades, and A was already assumed to be more likely.
But we don’t really know p(A|G), either for low, average, or high grade levels from the problem description, so I couldn’t actually say.
ETA: That should really be p(A|G!B), because, while A and B are independent variables, G is correlated with both. But I think everything still holds anyways.
4.) 35%; expected utility is p(A) times A, which leaves us with 1*10 and .35 times 35, or 10 and 12.somethingorother. We have expected returns of $12 for the 35% case, which is higher than the $10 case.
Why the answer is different: Because 1.C asks what are expectations are, and 1.B asks what the state of the class is
For b) and c), the questions were supposed to be the same—my bad, I have edited it. Please edit your answer accordingly.
Not all of your answers were correct (unsurprisingly, because I find some of the questions extremely hard—even I couldn’t answer them at first :D). I’ll wait for a few more replies and then I’ll post the correct answers plus explanations.
Oddly, my answers remained the same, but for different reasons. Also, I changed my answer to 1.D, and would recommend you change the wording to “Expected average” wherever you merely refer to the average.
1.A) Approximately. (Originally this was yes, until you stated that there were at least 700 million people on the planet. After that information, I updated this answer, because I realized that the problem had an additional assumption of a finite number of people, thus encountering any one left-handed person reduces the odds, very very marginally, of any different future person I encounter being left-handed, because the pool of people I’m drawing from now has slightly different odds.)
1.B) No. (Still.)
1.C) Approximately. Why the answer is different without resorting to math: In 1.B, we nonrandomly pull 10 right-handed students out of the group. In a pool of 24 10-sided die we’ve already rolled, we’ve pulled out 10 of them which did not roll 1; this does not alter the number which did roll 1, increasing their relative proportion. In this case, we’ve rolled the dice 10 times, and they never came up 1; the remaining 14 times remain fair dice rolls.
1.D) (Modified) Approximately.
1.E) Very very slightly.
2.) [Edited; apparently I screwed up when I added the possibility of an exact match] .41%, still assuming we’re not considering the proximity of 0 to 3, and including closer matches. (That is, only considering identical digit matches.)
3.ab) Supposing it’s more likely that a higher quality student is A than !A; it’s possible that it’s extremely unlikely for a person who isn’t high A to have high grades while still having more high grade students who aren’t A than are A, if the odds of A are substantially lower than the odds of being neither A nor B but still having high grades. So there’s not enough information.
Assuming it’s more likely you’re A and have high grades than ~A and have high grades, however, and assuming that this distribution holds for the grade average for each college (p(A|G) > .5 for all three G), you should in all cases favor low-B students, because the remaining pool of accepted students is more likely to be A than !A, because !B limits you to the pool of students who are either A or !A with high (enough) grades, and A was already assumed to be more likely.
But we don’t really know p(A|G), either for low, average, or high grade levels from the problem description, so I couldn’t actually say.
ETA: That should really be p(A|G!B), because, while A and B are independent variables, G is correlated with both. But I think everything still holds anyways.
4.) 35%; expected utility is p(A) times A, which leaves us with 1*10 and .35 times 35, or 10 and 12.somethingorother. We have expected returns of $12 for the 35% case, which is higher than the $10 case.
For b) and c), the questions were supposed to be the same—my bad, I have edited it. Please edit your answer accordingly.
Not all of your answers were correct (unsurprisingly, because I find some of the questions extremely hard—even I couldn’t answer them at first :D). I’ll wait for a few more replies and then I’ll post the correct answers plus explanations.
Oddly, my answers remained the same, but for different reasons. Also, I changed my answer to 1.D, and would recommend you change the wording to “Expected average” wherever you merely refer to the average.