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The problem is to find the number of elements in the sample space.

We have four patients and 4 categories of diseases.

Event A: All patients are in the same category

Event B: 2 patients belong to the same category

My question is why is the answer here $4^4$.

As far as I see it:

for one category (S...same category, D...different category):

A = {SSSS}

B ={SSDD, DSSD, DDSS, SDDS, SDSD, DSDS}

So why cant we just multiply 7 times 4 categories and get a sample space of 20?

VLC
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    First, $7\times 4 = 28\neq 20$... Next, the choice of sample space is a choice. The only rules per se are that you must do it in such a way that all events you are interested in referring to must be subsets of your sample space. Now... some sample spaces will be preferred for various reasons such as keeping the outcomes in the sample space equiprobable or aesthetically pleasing etc... – JMoravitz Oct 20 '21 at 18:21
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    Overall, all 4 patients could be in 4 possible categories each which is $4\times4\times4\times4 = 4^4$. With 4 categories of diseases, there are 4 ways all the patients could be in the same category. So sure it is true for Event A that the element SSSS could be replaced by disease category 1, 2, 3, or 4, and that is a straight multiplication, to 4 elements. But for Event B, there are 4 ways the category can be the same and for each of those 4 there are 3 ways the other category can be different. So now you can not simply multiply by 4. This is a combinatorics problem of $n\choose{r}$. – Gwendolyn Anderson Oct 20 '21 at 21:56

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