Find the probability that a word with 15 letters (selected from P,T,I,N) does not contain TINT If a word with 15 letters is formed at random using the letters P, T, I, N,
find the probability that it does not contain the sequence TINT.
(I just made up this problem.)

If "TINT" occurs in $k\gt0$ particular positions and there are $j$ overlaps between them, there are $\binom{k-1}j$ ways to choose the overlaps, $\binom{15-3k}{15-4k+j}$ ways to choose the positions and $4^{15-4k+j}$ ways to choose the remaining letters. Thus by inclusion-exclusion the desired probability is
$$
1+\sum_{k=1}^4(-1)^k\sum_{j=0}^{k-1}\binom{k-1}j\binom{15-3k}{15-4k+j}4^{j-4k}=\frac{1024574807}{1073741824}\approx95.42\%\;.
$$
