Joanne has a blank fair six-sided die and six stickers each displaying a different integer from 1 to 6. Joanne rolls the die and then places the sticker labeled 1 on the top face of the die. She then rolls the die again, places the sticker labeled 2 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 p 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 p can be written as nm, where m and n are relatively prime positive integers. Find m+n.
B: at the end of the process exactly one face has been left blank
E: all the even-numbered stickers are visible on faces of the die
By Toolkit 127 (Conditional Probability) P(B∣E)=P(E)P(B∩E)
E: after placing sticker 2, stickers 3, 4 cannot land on its face; after placing sticker 4, stickers 5, 6 cannot land on the faces of 2 or 4. P(E)=11×12×653×654×645×646=8125
B∩E exactly one face is blank (6 cases)
exactly one face contains two stickers (5 cases)
Since all even stickers must remain visible, the lower sticker must be odd.
