Probability puzzles

Solutions to all probability puzzles mobile app.

Catalan numbers $C_n$ is the number of monotonic lattice paths along the edges of a SQUARE grid with n × n square cells, which do not pass above the diagonal.

Bertrand ballot problem for p > q (RECTANGLE proble) probability of strictly in creasing win is $(p-q)/(p+q)$

Bounded Gamblers ruin

Unbounded Gamblers ruin

Assume probability $p$ to go up and probability $q$ to go down. Let $k_i$ be the expected hitting time of 0 from state $i$ and $h_{i}$ be the probability of hitting 0 from state $i$ at some point in time.

If $p < q$ then and $h_{i} = 1$ and $k_{i} = \frac{1}{q-p} * i$

If $p > q$ then $h_{i} < 1$ and hence $k_{i} = infinity$

If $p = q$ then $h_{i} = 1$ but $k_{i} = infinity$

When probability of hitting 0 is less than 1, then always mean hitting time is infinity as there is small chance to never hit 0 so mean hit time is infinity.

When probability of hitting 0 is 1, then mean hitting time can be infinity or not.