Back at twelve after twelve steps
An ant sits on the 12 of a clock face. It makes 12 steps; each step goes to one of the two neighbouring hour marks with probability .
Find the probability that the ant is on the 12 after the 12 steps.
Fractions, decimals with at least 3 significant digits, and expressions such as sqrt(2)/2, e^-1 or pi/4.
Track the net number of steps clockwise. Which values bring the ant back to 12?
Do not forget the walks that go all the way around the clock.
h
s Shown after a correct answer. Revealing it first marks the problem as seen, not solved.
Idea. The ant is back on 12 when its net clockwise displacement is a multiple of 12. With 12 steps of , that means a net displacement of 0, 12 or .
Step 1: net displacement 0. Six steps each way, in any order: walks.
Step 2: a full lap. All 12 steps clockwise, or all 12 counterclockwise: 2 walks.
Step 3. Out of equally likely walks,
The trap. Answering and forgetting the two full laps.