Need help with e part. Please

d) When I lived in Toronto, in the Fall I biked home from work in the dark. My route had 4 potholes on it and I hit all four each time. I'd start my commute with my bike light turned on, but it had a loose electrical connection so each time I hit a pothole with probability p the light would switch from on to off or from off to on. What was the probability that my light would be turned on after my commute? (2 points) Similar to the previous part, assume that the number of potholes varied and the probability that I hit k: potholes during my commute was A19 eXp[)\\)/k:!, for a xed A > 0. This is the probability distribution function of a Poisson. This distribution arises when potholes occur independently with rate A. Suppose that the probability that my light is switched (to off if on, to on if off) over each pothole was now p = 1. What was the probability that my light would be turned on after the commute, in terms of X? (As before, my commute began with the light on). You may express your answer using trigonometric or special functions: Do not leave it as an innite series