, 2021

Solution:

Let $p_{n}$ be the probability that the $n$$\text{th}$ head is flipped after a tail and Jude has yet to flip 3 heads consecutively to this point. For example, $p_{2}=\frac{2}{3}$, as it is impossible for 3 heads to be flipped consecutively and the second head comes after a tail exactly when the first flip after the first head is a tail, which happens with probability $\frac{2}{3}$. Similarly, $p_{3}=\frac{3}{4}$. We now establish a recursion between values of $p_{n}$:
$$p_{n}=\frac{n}{n+1} p_{n-1}+\frac{1}{n+1} p_{n-2}.$$
The first term comes from when the previous head had tails both before and after, and the second term comes from when the previous 2 heads were consecutive. Of course there cannot be other terms, as this would imply that 3 heads were flipped consecutively. This enables us to easily compute the next few terms: $\frac{11}{15}$, $\frac{53}{72}$, $\frac{103}{140}$, and so on. Notably, the differences between consecutive terms (starting from $p_{3}-p_{2}$) are $\frac{2}{24},-\frac{2}{120}, \frac{2}{720},-\frac{2}{5040}$, and so on. This leads us to guess that $p_{n}=2 \sum_{i=0}^{n+1} \frac{(-1)^{i}}{i!}$, which indeed satisfies the given recurrence relation. Then
$$\lim _{n \rightarrow \infty} p_{n}=2 \sum_{i=0}^{\infty} \frac{(-1)^{i}}{i!}=\frac{2}{e}$$
But since the probability that the $n$$\text{th}$ head comes after a tail approaches 1 as $n$ increases, this limit is the same as the limit of the probability that the first $n$ heads do not include 3 that came consecutively. Then this limit is just the probability that we never flip 3 consecutive heads. Then the desired probability is just $p=1-\frac{2}{e}$. We are asked to compute $\lfloor 180 p\rfloor$. This is the floor of $180-\frac{360}{e}$. To compute $360 / e$, note that we can just truncate the infinite sum
$$\frac{360}{e}=\sum_{n=0}^{\infty} \frac{360(-1)^{n}}{n!}$$
as it converges rather quickly. The first several terms are $360-360+180-60+15-3+\frac{1}{2}$, and the rest are insignificant. This sums to about 132.5, giving an answer of $\lfloor 180-132.5\rfloor=47$.