# No possibilities are ruled out

Suppose that two binary (yes-no) qualities are being considered. Often (yes, actually!) I want to express that all four combinations are possible: yes-yes, yes-no, no-yes, no-no. Is there a concise way to convey this?

I don't want to say that the two are independent, because they often are not: merely that no possibilities can be ruled out.

Example: A person is from Party A or Party B, and votes for or against Party A's bill. (The person might be on either side, and could either defect or vote with her party.)

Non-example: A number is negative or not, and the number plus one is negative or not. (The combination "x is not negative, but x+1 is negative" cannot occur.)

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You want a quick way to say that "no possibilities can be ruled out?" How about, "no possibilities can be ruled out?" – Evan Harper Aug 29 '12 at 5:47
@EvanHarper: I don't want to say that because of the large risk that it's understood as "None of {(x negative), (x nonnegative), (x+1 negative), (x+1 nonnegative)} can be ruled out" rather than "None of {(x negative, x+1 negative), (x negative, x+1 nonnegative), (x nonnegative, x+1 negative), (x nonnegative, x+1 nonnegative)} can be ruled out". Of course this example is false... – Charles Aug 29 '12 at 13:58

I don't fully grok the question. But, to answer the first bit:

Suppose that two binary (yes-no) qualities are being considered. Often (yes, actually!) I want to express that all four combinations are possible: yes-yes, yes-no, no-yes, no-no. Is there a concise way to convey this?

Depending on how mathematically accurate you want to be, you could use one of:

All permutations are possible.

All combinations are possible.

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I don't think permutations are applicable here, since order is not important. – Jim Aug 29 '12 at 5:54
@Jim Hence my note on mathematical accuracy. In my experience, even when inaccurate, permutations appears to be preferred over combinations when speaking about matters of choice. My guess is that this is because of the latter's more mainstream meaning of to join. – coleopterist Aug 29 '12 at 6:02

You might say:

"Given two binary values with non-zero probability for each combination"

Or

"Given two binary values whose Expected Value is a function of all combinations"

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The second would be true of my non-example as well. The first does not make the meaning of combinations clear (which is a real concern, since apparently it's difficult to describe the concept even in four paragraphs). – Charles Aug 29 '12 at 13:55