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.

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