maths help(probability)

Question

A well-traveled highway has its traffic lights green for 82% of the time. If a person traveling the road goes through 8 traffic intersections, find
a) the probability that the first
red light occur on the nth traffic light
b) the cumulative probability that the person will hit the
red light on or before the nth traffic light.

Answer 1

82% chance of a red light,

8 lights

chance of getting the 1st light red = .82

chance of getting the 2nd light red & 1st green or yellow = .82(.18) = .1476

chance of getting the 2nd light red and 1st light red = .82(.82) = .6714

chance of getting the 2nd light red, regardless of the 1st light = ..1476 + .6714 = .82

chance of getting the 1st, 2nd, 3rd, 4th, 5th, 6th, 7th, or 8th light red regardless of the other lights = .82

chance the 1st red light occurs on the 1st light = .82

chance 1st red occurs on 2nd light = .1476

chance 1st red on 3rd light = .82(.18^2) =.026568

chance 1st red on 4th = .82(.18^3) = .00478224

chance 1st red on 5th = .82(.18^4) = 8.608032 x 10^-4

chance 1st red on 6th = .82(.18^5) = 1.54944576 x 10^4

chance 1st red on 7th = .82(.18^6) = 2.78900237 x 10^-5

chance 1st red on 8th = .82(.18^7) = 5.02020426 x 10^-6

a) chance of red on nth light = .82(.18^(n-1))

b) chance of red on or before nth light = .82(.18^(n-1)) + .82(.18^(n-2)) + ... + .82

= .82(.18^(n-1) + .18^(n-2) + ... + 1)