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You are right that I shouldn't have used the word "odds" but that doesn't change the fact that as each person picks the chance of picking the purple rock changes for the next person.
Only the first person to pick has a 1 in 8 chance of picking the purple rock. Your chances change depending on when you get to pick & what the person(s) before you picked. So there definitely is a factorial if you want to figure the risk of pulling the purple rock for everyone before anyone draws and that risk is 1 in 5,040 (7factorial) because that is how many different ways the 8 picks can go.
If 1st person picks not purple rock, the chances of the 2nd person picking purple rock is now different (2nd person now has a 1 in 7 chance of picking the purple rock), etc.
And "7:1 against" is irrelevant as, again, this is only for the first person to draw.
The reason the probability of drawing the purple rock out of 8 rocks is a 1/7factorial before anyone draws is because you don't count the purple rock in the picks because then it would be 8 draws 8 rocks, which takes into account the draw that picks the purple rock (i.e., if first person draws purple rock then the other 7 have 0% chance of drawing the purple rock; no need to count that draw). You can write it as 8/8factorial, but 8/8 is 1 & since the denominator is a factorial it is 1/1x7x6x5x4x3x2x1 or 1 in 5040.
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