Ant crawls along a number line

An ant is crawling on a number line. The ant starts out at position \(0\). Every second the ant either

  • Moves to the right 1 unit, with probability 1/2,
  • Moves to the left 1 unit, with probability 1/4, or
  • Stays at its current location, with probability 1/4

The ant’s movement during a particular second is independent of the ant’s previous movements. Let \(X_{160}\) be the ant’s location after 160 seconds.

  1. In 160 seconds, what is the probability that the ant moves to the right exactly 80 times, and to the left exactly 40 times?
  2. What is \(\mathbb{E}(X_{160})\)?
  3. What is \(Var(X_{160})\)?
  4. Let \(\mu=\mathbb{E}(X)\). Estimate \(\mathbb{P}(|X-\mu|\geq 15)\).

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