Joanne has a blank fair six-sided die and six stickers each displaying a different integer from to . Joanne rolls the die and then places the sticker labeled on the top face of the die. She then rolls the die again, places the sticker labeled on the top face, and continues this process to place the rest of the stickers in order. If the die ever lands with a sticker already on its top face, the new sticker is placed to cover the old sticker. Let be the conditional probability that at the end of the process exactly one face has been left blank, given that all the even-numbered stickers are visible on faces of the die. Then can be written as , where and are relatively prime positive integers. Find .
Let be the event that exactly one face has been left blank, and let be the event that all of the even-numbered stickers are visible. Let the outcomes of the dice rolls be denoted .
First, note that occurs if and only if (the fourth sticker isn't replaced) and (the second sticker isn't replaced). Hence:
where the first product comes from making sure and are distinct from and the second product comes from making sure and are distinct from .
Note that the event is equivalent to there being exactly one repeat in the sequence . To find , we must take casework on which satisfy . Note that . Hence, there are possible ordered pairs, namely:
Once we pick our ordered pair , we have ways to assign it a number, and ways to assign the remainining numbers, so there are a total of possible sequences. Hence:
and so:
Hence, the answer is .
The problems on this page are the property of the MAA's American Mathematics Competitions