Okay, but ... ack. The logic still doesn't seem very logical to me. I think would have needed the question to be worded, "If and only if Jenny hits a home run, her team will win" for me to get the answer, because otherwise I start overthinking. Is the "If this is true" part supposed to serve the same function as "if and only if" would in my wording?
I think you're ignoring the "will" -- and "is true" the guarantee part of the question as worded.
If
Jenny hits a home run, her team
will
win.
Given
that this
is true,
what else also must be true?
It's not an if and only if question. They are not stating Jenny's hitting a homer is The Only Way to win. That would lead to different answers. That's a different question. They're giving you the condition, and the definite outcome if that condition is met. It is conditonal she will hit the run. But if that condition is met, her team will win (that's the guarantee).
Here's some examples, LJ.
One way to look at it is that "if Jenny hits a home run" can also mean "if Jenny crosses home plate." Under either circumstance, the team will win. So:
a. If the team won, Jenny hit a home run.
No, because Jenny could've hit a single, double, triple, or walked, etc., and then scored later. Not hitting a home run does not necessarily mean you are out (except on Home Run Derby).
b. If Jenny didn't hit a home run, the team tied.
No, because we have no idea what the score is, or how many runners are on base.
c. If the team didn't win, Jenny didn't hit a home run.
Yes, since we know they would've won no matter what if she did hit a home run, then she must not have hit one.
"If X then Y" is congruent to "If not-Y then not-X" in formal logic. or, because I like symbolic logic, X->Y = ~Y->~X.
If the team did not win, she couldn't have hit a home run, because if she had hit a home run, the team would have won.
If Jenny doesn't hit a home run her team will be so discouraged they will quit and forfeit. OK, maybe not helpful.
Trust me, I'm not getting confused by an excess of baseball knowledge on this one.
So "if X then Y" is the same as "if not Y, then not X" - Y is the bigger group, it can include X inside it.
If Nilly says this is true-enough-for-such-games, even if not strictly true all the time, that is good enough for me. I kept thinking it wasn't true, which is what screwed with me. (I always turn word problems into Xs and Ys, because I used to get more confused if I was thinking about specifics. Apparently, this is a case where that was not helpful to me.)
they would've won no matter what if she did hit a home run, then she must not have hit one.
This actually makes it make more sense to me.
Okay, thanks. Except I feel dumb now.
Last week I received an example of the kind of thing that may have made Lyra suspicious of the logic of answer C in the real world. It was in a newsletter from State Farm Insurance. The headline asked, "Is your car at increased risk of being stolen?" Then they gave a list of "high risk" cars, which, surprisingly, did not include Porsches or Corvettes but instead the Camry, Accord and Ford pickup truck.
Obviously, the data that they had were that if a car is stolen, it is more likely to be fall into this group of cars. This is hardly suprising because there are more of these popular cars in every city in America than there are Porsches or Corvettes in the whole country. They twisted this conditional probability P(car type/stolen) around to the opposite conditional probability: if a car falls into this group, it is more likely to be stolen P(stolen/car type). There is no necessary equivalence between P(A/B) and P(B/A), but the human brain seems to be prepared to treat them as equivalent.
I can only hope that it was State Farm's marketing group who came up with this, not their actuaries.
the only thing missing from that CD is "alarm you left on while gone on vacation" which might be the most annoying neighbor noise I ever had.
I also want one of these.
