Question for probability theorists out there. On my logic quiz this week, the students had to guess/figure out which of 9 arguments were valid. As it turned out, 5 of them were valid. Though surprisingly many thought that All philosophers are academics, no academics are ghosts, so some ghosts are not philosophers was also valid. I thought this was a pretty good example of why you don’t want Aristotle’s version of existential import. I also thought that examples of arguments that actually have true premises and false conclusions would be softball questions. Not so it seems.

The scoring worked as follows. For each of the 5 valid arguments that the students recognised as such, they got 2 points, for a maximum of 10. For each of the 4 invalid arguments they marked as valid, they were deducted 2 points, with the exception that the lowest possible score is 0. So a student who said that 1 of the valid arguments and 3 of the invalid arguments were valid would get a score of 0, not of -4.

Now let’s assume that a student does just guess. (I hope this was true of some of them.) What is his (or her) expected score? If I don’t have the zero minimum rule, this is an easy question. The answer is 1. But with the minimum in, it does not seem so easy. I could just compute all 512 possible answers and add up the scores from them, but that would be tedious. Is there an easy way to compute this?