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I've been recently studying probability and I'm pretty much stuck with analyzing the difference between finding sample space of rolling a pair of dice simultaneously and drawing two cards simultaneously from a pack of 10 cards each card numbered 1 to 10. In the first case, the sample space is 36 that is we consider all the possible pairs (for example the pairs (1, 2) and (2, 1)) are treated separately. But in the second case the sample space is 10C2 (that is here the pairs are considered the same). My doubt is that in the first case since the dice are rolled simultaneously why can't we consider the pairs as the same which when considered the sample space for first case becomes 21 but not 36?

Bhavesh Pendyala
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    Well..the two cards can not be chosen independently. They can't match, for example. – lulu Feb 24 '23 at 14:38
  • Not sure what you are asking. You can, of course, consider either ordered or unordered pairs of dice throws (or ordered and unordered pairs of cards). There's no right or wrong there, though either one or the other might be better suited to whatever you are trying to describe. Note, however, that if you do consider, say, unordered dice throws, that the probabilities are not uniform. You are twice as likely to roll ${1,2}$ as you are to roll ${1,1}$. Again, there's nothing wrong with that, but it is a common source of error. – lulu Feb 24 '23 at 14:44
  • When you use counting arguments, you want each possibility to be equally likely. With your cards (without replacement) it may not matter whether you count $55$ unordered pairs or $110$ ordered pairs, but with your dice (with replacement) the $21$ unordered pairs are not equally likely while the $36$ ordered pairs will be with fair dice – Henry Feb 24 '23 at 14:48

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