medium Probability
A release-checked medium problem for training Law of Total Probability, Conditional Probability, Markov Chains, Recurrence Relations.
Consider a particle that performs a random walk on the integers starting at position $0$. At each step, the particle moves from position $i$ to position $i+1$ with probability $p$, while it moves from position $i$ to $i-1$ with probability $1-p$. If $p = 3/4$, find the probability the particle ever reaches position $1$.
This Probability problem is tagged Law of Total Probability, Conditional Probability, Markov Chains, Recurrence Relations. State the random variables and conditioning information explicitly, then check the result against boundary cases before opening hints or a solution.
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