# & % Question

[sparrrkk_]

Hi,

I have 2 examples of #% questions that I wanted to clarify since they keep confusing me.
1. 20% of girls have a dog. 30% of boys have a dog. Therefore, boys are more likely to have a dog than girls. (what is the flaw?)

I was hoping you could clarify what "more likely" means. I know that it refers to a correlation but I'm confused as to whether it means a % and if so, which %?

2. Right-handed people are more likely to cause accidents than left-handed people. So, there are more right-handed people causing accidents than left-handed people.

I understand that a flaw of this reasoning would be that it doesn't account for when there is a small # of right-handed people and a big # of left-handed people. For example, if there were 10 right-handed people and 100 left-handed people, 50% of 10 would be 5 right-handed people who cause accidents vs. 10% of 100 left-handed people would be 10 left-handed people who cause accidents. Thus, there would be less right-handed people causing accidents than left-handed people.

For some reason, I can't reconcile the "more likely" usage in both examples to mean the same thing...

As you can probably tell, I am very confused so I would super appreciate any help! Please break down each example and provide a definition for "more likely". Thank you so much!

[Robert Carroll]

spark,

I think your first example is a bit ambiguous, and would benefit from some more context. Take the premises to be as you wrote them, so:

Premise 1: 20% of girls have a dog

Premise 2: 30% of boys have a dog

One way to interpret the conclusion is as follows:

Conclusion: Boys are more likely to have dogs than girls are.

Under this interpretation, I really don't see any flaw in the argument. An arbitrary boy would be 30% likely to have a dog, and an arbitrary girl would be 20% likely to have one. That seems to validate the conclusion.

Try a different interpretation of the conclusion:

Conclusion: A dog is more likely to be owned by a boy than by a girl.

This interpretation doesn't follow from the premises, because it may be that they are so many more girls than boys that more girls than boys own dogs, even if a smaller percent of girls own a dog. So most dogs could be owned by girls, not boys, making the likelihood higher for girls. Of course, other groups could own dogs (women, men), but even restricting the analysis to boys and girls, the conclusion doesn't have to follow.

I think that actually the first interpretation fits the statement better, so this isn't a flawed argument. Can you tell me where you got the example from?

As for the second example, I think the premises suffer a possible ambiguity about what "more likely" means. Again, can you tell me where the examples are from?

I will follow up with examples of my own that are unambiguous!

Robert Carroll

[Robert Carroll]

spark,

Here's an example corresponding to your #1:

30% of boys in Whoville own a dog. 20% of girls in Whoville own a dog. No one in Whoville owns more than one dog, and no dog in Whoville is owned by more than one person. The lost dog we found in Whoville is thus more likely to belong to a boy than to a girl.

This is a clear example of a flaw that makes an assumption about the numbers of boys and girls. In order to know whether the lost dog "more likely" belongs to one kind of resident versus another, we would need to know how many boys and girls there are in order to see how many dogs girls own versus how many dogs boys own. I think the argument may make other assumptions, but those are in addition to that assumption about numbers.

An example corresponding to your #2:

The likelihood that a right-handed resident of Whoville will cause an accident is higher than the likelihood that a left-handed Whoville resident will cause an accident. Therefore, there are more accidents in Whoville caused by right-handed than by left-handed residents.

This has the problem you identify in your second example.

Robert Carroll

[sparrrkk_]

Thank you so much! This really clarified this confusing topic for me. As for those examples, I wrote them down a long time ago and forgot to write down which problem it came from, so I don’t know where I got them from. However, your examples were really clear, so I think I got it! I’ll review this some more and follow up if I have more questions.

[sparrrkk_]

So, just to clarify, with the second example regarding "A is more likely to cause accidents than B. There are more A causing accidents than B". A possible way to weaken this argument is by considering the situation where there are less total A than total B. Then, even though A has a higher %, it has a lower actual #.

[Adam Tyson]

You got it! In many problems involving numbers and percentages you will see two groups being compared. If the groups are the same size, the argument usually makes sense, so you should consider what happens if the two groups are of very different sizes.