Series of 10 events probability?

[MathsNoop]

Hello!

I have a 10 stage series of events, and I'm looking for confirmation of my maths, as I'm not 100% (1, 1/1, 1.0) I'm correct.

Each stage has a 25% chance to proceed to the next, and a 75% chance to reset to zero. The first stage is always 100%.

Am I correct in calculating that;

0-1 = 100%
0-2 = 25%
0-3 = 6.25%
0-4 = 1.56%
0-5 = 0.39%
0-6 = 0.097%
0-7 = 0.024%
0-8 = 0.0061%
0-9 = 0.0015%
0-10 = 0.00038%

Additionally can anyone let me know the average amount of times this series would need to be ran to achieve each stage, as a fraction?

Thanks in advance!

 

[JeffM]

Your computations look accurate (I did not check them all).

Your follow up question is unclear. You can never achieve certainty with a probabilistic process. So in principle you could run the process forever without getting to stage 10.

Let’s be sure we understand the process. There are actually eleven stages, numbered zero through ten. In stage zero, the probability is 1 that on the next try you proceed to stage one. In stages one through nine, the probability is 25% that on the next try you advance to the next stage and 75% that you revert to stage zero. At stage ten, the process stops. Is that correct?

So is your question, what is the expected number tries required to get to stage 2, 3, 4, etc?

 

[MathsNoop]

All correct yes.

 

[Dr.Peterson]

Hello!

I have a 10 stage series of events, and I'm looking for confirmation of my maths, as I'm not 100% (1, 1/1, 1.0) I'm correct.

Each stage has a 25% chance to proceed to the next, and a 75% chance to reset to zero. The first stage is always 100%.

Additionally can anyone let me know the average amount of times this series would need to be ran to achieve each stage, as a fraction?

As I understand it, you want to calculate the expected value of the number of attempts required to achieve a particular stage. I'm not sure what you mean by "as a fraction". Can you clarify that?

But if I'm interpreting your request correct, you should first calculate the probability of taking x attempts to reach stage N (that is, attaining that stage on the xth attempt, and not before).

Start with N=1. It appears that you are starting at stage 1, so it takes 0 attempts to reach stage 1 with P(x) = 1 (that is, 100%). Am I right?

Now do the calculation for N=2. What is P(0)? P(1)? P(2)? This will help in understanding the larger problem. (I haven't yet tried doing this.)

Please show your work, so we can see where you need help.