A frog starts at “0” on a number line. Each second, it flips a fair coin,
and moves forward 1 unit if the coin shows heads, and forward 3
units if the coin shows tails. What is the probability that the frog will
eventually land on “8?” Express your answer as a common fraction.
What is the number of solutions to h + 3t = 8?
What is the probability that the frog goes over '8'? What happens if it gets three 3s?
Can you use modular arithmetic?
h+3t = 8
h = 2 (mod 3)
Let h be the number of heads.
Let t be the number of tails.
h = 3n + 2
so, t = 2-n
Ordered pairs of solutions:
(2, 2) ~ (5, 1) ~ (8, 0)
Non solutions that are greater than 8 but no extra leap over:
(0, 3) ~ (1, 3) ... count the rest, being mindful not to go an extra leap after you have reached >8.
Then answer is 3/ (non solutions + 3 )