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I have an issue where I want to see if two events may coincide by more than chance.

For instance given the following data I may hypothesise that "Alice" and "Bob" come into the office together more frequently than chance.

IN_OFFICE   ALICE    BOB    CHARLIE  EVE
MONDAY      x        x               x 
TUESDAY     x        x      x        
WEDNESDAY                            
THURSDAY    x               
FRIDAY
SATURDAY    x        x      x        x 
SUNDAY      x

I know the chance of Alice being at work on a day:

  • $P(A) = 5/7$

And I know chance of Bob being at work on a day:

  • $P(B) = 3/7$

And I know the "observed" chance of Alice and Bob being both at work on a day:

  • $P(AB) = 3/7$

And I can calculate my null hypothesis "expected" chance that Alice and Bob are at work on a day

  • $E(AB) = P(A) \cdot P(B) = 5/7 \cdot 3/7 = 15/49 \approx 2/7$

And I can see that "observed" is higher than "expected".

I'd assumed that was some sort of $\chi^{2}$ test, but my data is boolean (in office or not in office) rather than discrete, and I get odd results back when I try it.

Is this along the right lines? Is there some way of putting a $p$ value (or similar) on this, to say whether this difference is significant or not? And does the presence of Charlie and Eve affect the results?

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