Solution
No Two Heads Adjacent
Show the problem again
A fair coin is flipped 10 times. What is the probability that no two heads occur consecutively? Express as a fraction.
Worked solution
The answer is 144/1024 = 9/64. Let a_n count length-n binary strings with no two adjacent H. A valid string ends in T (preceded by any valid string of length n−1) or in TH (preceded by any valid string of length n−2), so a_n = a_{n−1} + a_{n−2} with a_1 = 2, a_2 = 3. These are Fibonacci numbers, a_n = F_{n+2}. So a_10 = F_12 = 144 and the probability is 144/2¹⁰ = 9/64.
Source: Standard Fibonacci-flavoured coin-sequence exercise from probability texts. Statement written for AxiomIQ.