Q3 - For any given ticket in a 1000-ticket lottery, it is re

[goriano]

Could you explain why (B) is wrong? I thought it mirrored the flaw by going from THAT ticket/THAT horse to ALL tickets/ALL horses?

[ohthatpatrick]

Could you explain why (B) is wrong? I thought it mirrored the flaw by going from THAT ticket/THAT horse to ALL tickets/ALL horses?

You're right that (B) does go from a specific premise to a general conclusion.

However, the flaw in the original is taking one trait [reasonable to believe that it will lose] and saying, since that trait applies to each individual ticket, we may conclude that that trait applies to all tickets.

In (B), the trait that applies to the specific premise is [reasonable to believe that it will win], while the trait applied in the conclusion is [reasonable to believe it will not win].

Here's another distinction:
In the original argument, the specific thing discussed in the premise is a subset of the larger group discussed in the conclusion.

In (B), the specific horse discussed in the premise is NOT a subset of the larger group discussed in the conclusion (the conclusion is about the group of all horses other than that premise-horse).

FYI, on Matching questions, beware the "Topic Trap" - answer choices that seem to be very close in topic to the original argument. (The test wants matching logic, not matching topics)

I would be dubious of (B), (C), and (D) on this question because they seem to be baiting me into thinking the number 1000 is critical to the flaw.

There's actually a common name for the recurring flaw found in the original: Part to Whole.

This flaw occurs when an author assumes that some trait that applies to a certain part therefore applies to the whole.

This flaw also works in reverse, in which case we often call it Whole to Part.

Here are a couple examples:
Each member of this new committee is an efficient worker. Thus, this will be an efficient committee. (part to whole)

It is wrong for an individual to have power over government, so therefore it is wrong for society, a collection of individuals, to have power over government. (part to whole)

Collectively, the pillars on this building are enough to support the weight of the roof. Therefore, each pillar on its own could support the weight of the roof. (whole to part)

The telltale sign of this flaw is that the same trait is discussed in the premise and in the conclusion.

====other answers====

A) is correct. Since any given card will probably not be an ace, the author thinks we can conclude that all cards will probably never be an ace.

C) is not a part to whole flaw. It's a flaw that's more like "because it's unreasonable to believe that A will happen, it's reasonable to believe that A will never happen"

D) This is not really a flawed argument. The first sentence is, in reality, a flawed notion of probability, but if we accepted the first sentence as true, then we would believe the logic of the conclusion.

E) is the flaw of reversed conditional logic (sometimes called nec/suff conflation). They give us 5 yrs. old --> prob 1 meter tall, and then argue that since someone is 1 meter tall, she is probably 5 yrs. old.

[agersh144]

Great explanation and outstanding examples -- well done!

[WaltGrace1983]

E) is the flaw of reversed conditional logic (sometimes called nec/suff conflation). They give us 5 yrs. old --> prob 1 meter tall, and then argue that since someone is 1 meter tall, she is probably 5 yrs. old.

Unless I am missing something here, isn't (E) flawed because it goes from "whole" to "part?"

(E) says that a group of five-year-old children = average height is one meter. Therefore, one five-year-old child = exactly one meter tall.

Also, can anyone give me a bit more info on (B)? I read the explanation above (thanks Patrick!) but I am not so sure that I got it completely. Here was my reasoning:

The original argument discusses the piece = puzzle issue. Just because something applies to a piece of the puzzle doesn't mean it applies to the whole puzzle. (A) matches this perfectly: it is reasonable to believe that a random card will not be an ace, thus it is reasonable to believe that no card will be an ace. In other words, it takes the piece (the ace) and applies it to the puzzle (the deck). We all know that it is NOT reasonable to believe that no aces will be drawn just because it is reasonable to believe that an ace will be.

As for (B), to me it has a very similar structure with a few differences that made me eliminate it. For one, (B) is talking about certainty. It is saying that it is reasonable to believe that X will happen when X happening is an almost-certainty (999/1000 chance). However, it goes on to conclude that this thing that was almost-certain is actually reasonably believed to be certain. To me, this parallels (A)'s reasoning very well but it flips it around. While (A) deals with a similar level of certainty (aka that this occurrence will probably not happen), (B) deals with an inverted level of certainly (aka that this occurrence will probably happen).

I couldn't eliminate it for any other reason and I am not sure, as I said, that I understand the explanation above totally. Thanks!

[ohthatpatrick]

You're totally right about (E). I bricked that. Good catch.

The flaw in (E) is really interpreting AVERAGE incorrectly (assuming that if a group AVERAGES a certain number, then every member IS that number).

That's super close to WHOLE to PART, although a traditional whole to part would sound like

since THE GROUP of five year olds is one meter high
then THIS five year old is one meter high.

But it's really saying "the average of any GROUP is one-meter high". So it's a tiny bit different.

(p.s. incidentally, the only way it could ever be true that "for ANY given group of 5 year old children, their average is 1" would be for every single member of the data set to have a value of 1 ... so technically, if we accept E's premise, then it is a totally valid mathematical conclusion to draw)

I find it helpful with PART/WHOLE stuff to force myself to spell out which trait I am saying applies to both.

since THE GROUP one meter
then THIS INDIVIDUAL PAT one meter

So for the original argument, we had

since ANY GIVEN TICKET no win
then ALL TICKETS no win

For (B), we can't transfer the same trait/adjective from Part to Whole.

since THIS HORSE will win
then ALL OTHER HORSES no win

So, it's not a Part/Whole unless you assign the same trait to both the part and the whole.

And then the other needed ingredient of Part/Whole is that the Part we're talking about is a member of the Group we're talking about.

The original talks about the GROUP of these 1000 lottery tickets in the conclusion. The premise talks about ANY GIVEN MEMBER of these 1000 lottery tickets.

(B) talks about THE HUGELY-FAVORED HORSE and then the GROUP of all the other horses.

The hugely-favored horse isn't part of that group.

I will say the one other thing that helps me get rid of (B) is that I actually find (B) to be pretty legit. As you pointed out, it's not LITERALLY fair to say that a 99.9% probability is reasonably certain, but it's pretty close! I'd definitely treat a 99.9% probability as though it were certain.

Whereas, I would never say "No lottery ticket will win ... or No cards in this deck are an ace!"

Hope this helps.

[WaltGrace1983]

I think I see what you are saying but I don't really understand where one can reasonably show that there are two distinct groups. It is probably just a shortcoming in my understanding. Let me try and reason this out and let me know if I got it if you could!

We are talking about a "certain horse" just like we are talking about a "certain card," an ace or a "certain ticket"! I'd assume that the problem is NOT with this statement. Therefore, the problem lies with the phrase, "So it is reasonable to believe that no one other than that horse can win." It seems that there might be somewhat of a comparison going on. It seems that IF (certain horse), THEN win. IF ~(certain horse), THEN ~win. There is separation.

In the original argument, the (ticket) is a part of a larger group, the (lottery). There is no separation between them. The (ticket) is a PART of the (lottery) and the (lottery) consists of the (ticket). Is this difference what you are getting at?

[ohthatpatrick]

Yeah, you nailed it.

The conclusion is about the NOT-group, i.e. NOT-this certain horse.

In the original, the conclusion is about ANY ticket, which includes the certain ticket we were talking about.

If I say, "of the 10 racers running a race, racer X is almost guaranteed to win. Thus, the other 9 racers are guaranteed to lose", then that's not a part to whole issue, because racer X is not part of "the other 9 racers". It's just a flaw involving going from "almost guaranteed" to "guaranteed".

Whereas the original is saying, "each one of these 1000 tickets is unlikely to win. Therefore the entire group of 1000 tickets is unlikely to possess a winner."

[krishna.kilambi]

Is option C a valid conclusion ? What is the flaw in the argument of option C?

[tommywallach]

Patrick answered that above.