Help determining chances of winning in a modified game of roulette

Question

Hey all. Wondering if you can help me out with the chances of winning in a round of N spins in a modified game of roulette

THE GAME

15 numbers, red/black and green

0 : GREEN
1-7 : RED
8-14 : BLACK

Chances of getting red/black = (7/15)
Chances of getting green = (1/15)

THE STATEGY

To bet one color (red/black) until you get that color, then switch to the other color

The first thing I want to calculate is the chance of getting to spin N and winning on spin N. Say for this example N = {1,2,3}.
I hope this makes sense, but I think that the chance of getting to and winning on spin N should be (the chance of getting to spin N) * (the chance of winning on spin N)

The chance of getting to spin N = (8/15)^(n-1)
The chance of winning on spin N once you are on spin N = (7/15)
So the chance of getting to and winning on spin N = (8/15)^(n-1)*(7/15)

S1 = The chance of winning on the first spin = 1*(7/15) = 0.46666667
S2 = The chance of winning on the second spin = (8/15)*(7/15) = 0.248888888
S3 = The chance of winning on the third spin = (8/15)^2*(7/15) = 0.132740741

Please correct me if this is wrong

So the next thing I'm trying to figure out is the chance of you winning in a round of N spins. So for this example N = 3. This is where I am unsure how to calculate your chance of winning. Do you simply just add up the chances?

So your chance of winning in 3 spins would simple be S1 + S2 + S3 = 0.848296291 = 84.83%

Does this all make sense?

Thanks in advance

Answer 1

If you are trying to calculate the probability of winning at least once in the first three spins then your answer is correct . You add because those 3 are mutually exclusive.

 

Remember for each Sn you are actually calculating the probability that your first win is on the nth (Sn) spin. It does not exclude the possibility that you win again after the nth spin; i.e S1 includes the possibility of winning on the second and/or third spin.

 

Nice work.

 

What course is this for?