Struggling with this Multiple Choice Question

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Smalls

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Teacher here. I created a probability test for my Algebra 2 students, and used this question from my item bank. But when I do the problem myself, I'm struggling to reach the right answer. Can you help me figure out where my mistake is?

A survey of high school seniors found that 45% take Statistics, 54% participate in a sport, and 39% do both. What is the probability that a randomly selected student does not take statistics or participate in a sport?

A0.38
xB0.46
C0.54
D0.84

If I'm reading the question right, it's asking for the P(not A or B) where A is taking statistics and B is participates in a sport. So P(not A or B) = P(not A) since we really only care that they aren't taking statistics? Or am I way off base here?

Thanks in advance for your help!
 
Yer killin me Smalls! (sorry had to do it)

First we need to make sure we agree on the interpretation of "does not take statistics or participate in a sport"
I interpret this as the student does not take statistics and the student also does not participate in a sport.
This would be NOT(statistics OR sport)

Let
[MATH]A: \text{Student takes statistics}\\ B: \text{Student participates in a sport}[/MATH]
[MATH]P[!(A \cup B)] = 1-P[A \cup B] = 1-(P[A]+P[ B ] - P[A \cap B]) =1- (0.45 + 0.54 - 0.39) = 0.4[/MATH]
 
Romsek,

I'll never forgive you for this! (Well, ok I will because everybody does it!)

That's what I thought originally, and is also the answer that I was arriving at. The only way that I was getting to their answer was if the numbers for statistics and sports were switched, and the question was asking what I described above.
 
I died along time ago from posts like this.

First of all I interpret the question the same as Romsek.

Now I am not going against that but 0.4 is not one of the choices.

So possible the author meant, as you say, not A or B. I will write not A as A'

Now, as you should know, ( A' or B) = A' only in the case if B is a subset of A'. That is every student in B is in A'. That means that every student who participates in sports is not taking statistics. But this simply is not true as it clearly says that 39% of the students do both. So P(not A or B) = P(not A) is not correct!

P(A' or B) = P(A') + P(B) - P(A' and B) = (.55 + .54 - ?). You figure out what P(A' and B) equals. For the record, the final answer is not one of the choices.

I want to state your error again. (X or Y) = (X union Y) = X if and only if Y is a subset of X, that is every element of Y is in X. In your problem that is not true with A' and B.
 
Out of curiosity why do you say P(not A or B) = P(not A) since we really only care that they aren't taking statistics? Why do we only care that they are not taking statistics?

Think about this carefully. You are saying that you can get different answers for P(not A or B) based on what we really care about. That is total garbage, as P(not A or B) has a unique answer, not multiple answers.

Are you really teaching probability using the above logic? That is terrible. We need to put our students first and that includes giving the students a teacher that has some idea of what is going on.

I know what I said is harsh but after re-reading your comment about the answer depends on what we really wants is just so outrages. It is one thing not to be able to do a problem or even to do it wrong but to say what you said is just unacceptable.
 
I didn't realize those were answer choices. durrr

If we take the statement to mean [MATH]!\text{taking statistics} \cup \text{participating in a sport}[/MATH]
Then what we have is

[MATH] P[!\text{statistics $\cup$ sport}] = P[!\text{statistics}] + P[\text{participating in a sport}] - P[\text{!statistics and participating in a sport}] \\ P[!\text{statistics}]=1-0.45 = 0.55\\ P[\text{participating in a sport}] = 0.54\\ P[\text{!statistics $\cap$ sport}] = P[\text{sport}]-P[\text{statistics $\cap$ sport}] = 0.54-0.39 = 0.15\\ P[!\text{statistics $\cup$ sport}] = 0.55 + 0.54 - 0.15 = 0.94 [/MATH]
So I don't know how that expression is to be interpreted if we are to match up with the choices.
 
I learned long ago that the English phrase "not A or B" is ambiguous. It may mean

not(A or B) = not A and not B = neither A nor B.

It may mean

either B or else (not A).

It may mean

B, or not A, or B and not A.

The problem lies in English usage. The English "or" (by itself) is a non-exclusive or like the Latin "vel." Some people use the "either ... or [else]" as an exclusive "or" like the Latin "aut ... aut," but that usage is far from universal. Moreover, the logical basis of not(A or B] equating to not A and not B is commonly expressed with "neither ... nor," which means but does not say "not ... and not...." Finally, does the "or" limit the range of "not," or does it not. The latter problem can be cured by placing positives before negatives.

Because this is your problem, I suggest that you decide what you want the problem to mean and then use

neither A nor B, meaning not AT AND not B; or

either B or else not A, meaning B or not A but not both; or

B or not A meaning B, not A, or both B and not A.
 
Jomo said:
Out of curiosity why do you say P(not A or B) = P(not A) since we really only care that they aren't taking statistics? Why do we only care that they are not taking statistics?

Think about this carefully. You are saying that you can get different answers for P(not A or B) based on what we really care about. That is total garbage, as P(not A or B) has a unique answer, not multiple answers.

Are you really teaching probability using the above logic? That is terrible. We need to put our students first and that includes giving the students a teacher that has some idea of what is going on.

I know what I said is harsh but after re-reading your comment about the answer depends on what we really wants is just so outrages. It is one thing not to be able to do a problem or even to do it wrong but to say what you said is just unacceptable.
Click to expand...
Unfortunately, I disagree. English syntax is very difficult with respect to expressing these logical distinctions exactly. The answer obviously depends only on what the problem is, but what the problem is can be known only through the words used to describe it. But if the wording is ambiguous, it is impossible to know what problem is intended and therefore impossible to know what the correct answer is.

In my previous post, I tried to elucidate the issues in English usage and how to avoid ambiguity in posing word problems on this topic. Of course, in real life, problems are all too often presented in fuzzy verbiage, and the only way to get clarity is through discussion. But you cannot have any discussion with a text book or computer screen, and consequently people who write problems need to understand these subtleties of English usage.
 
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Smalls said:
A survey of high school seniors found that 45% take Statistics, 54% participate in a sport, and 39% do both. What is the probability that a randomly selected student does not take statistics or participate in a sport?

A0.38
xB0.46
C0.54
D0.84

If I'm reading the question right, it's asking for the P(not A or B) where A is taking statistics and B is participates in a sport. So P(not A or B) = P(not A) since we really only care that they aren't taking statistics? Or am I way off base here?
Click to expand...
I think it's clear that this is a bad problem all around. It's not your mistake, but theirs. I think the confusion in your last paragraph is probably a result of trying to make their answer right, which would mess up anyone's mind.

I can't think of any reasonable interpretation of the question for which any of the answers is correct, so they're wrong at least in the list of choices. If you want to assign the problem, at least put 0.40 in the list.

Although the wording of "does not take statistics or participate in a sport" could be improved to make sure students are being tested only on math and not on grammar, I think it is clear that it means P(~(A U B)). If they meant ~A U B, they would have to say "does not take statistics, or participates in a sport". (The comma is recommended, the "s" is required.) Given that both verbs are infinitives following "does not", it has to mean "neither takes statistics nor participates in a sport" (which would be my recommended way of clarifying it). But good communication means making sure what you say can't reasonably be misunderstood, even under pressure, and this problem fails the test.

You might want to ask the publisher about it. I'm sure they get plenty of error reports.
 
Thank you guys so much for your responses! I really appreciate the discussion, and I'm glad to see I'm not the only one frustrated by the grammar of this question!
 
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