Biased random walk
= Biased random walk
{title2=$\mathbb E S_n=n(2p-1),\quad p\ne\tfrac12$}
A biased random walk on the <integer> line takes unit steps right and left with unequal <probabilities> $p$ and $1-p$, using <independent and identically distributed random variables> as <increments>. For $S_0=0$, the <expected value> is $n(2p-1)$: positive for $p>1/2$ and negative for $p<1/2$. Changing $p$ to $1-p$ reverses the preferred direction. The exact endpoint <probability mass function> is given under <simple random walk on the integer line>.
For example, with $p=0.65$ each step has <expected value> $0.30$ and <variance> $0.91$. After $80$ steps,
$$
\mathbb E S_{80}=24,\qquad \operatorname{Var}(S_{80})=72.8,\qquad
\sigma_{80}=\sqrt{72.8}\approx8.532.
$$
The displacement of the <expected value> from zero is about $24/8.532\approx2.813$ <standard deviations>, yet individual <sample paths> still move left as well as right, as the figure under <simple random walk> illustrates. Bias lets this elementary <stochastic process> model an average directional tendency while retaining fluctuations. Boundary and return questions have their own treatments under <gambler's ruin> and the specialized children below.