Every second, a green marble has a chance to appear. You are not told what the probability of this occurring is - instead, you are given a number which is equal to the average number of seconds it takes for a green marble to appear (let's call it 'mean time to happen').

Now let's say you want there to be a chance of a red marble appearing instead of a green one. In order to make this happen, you must define a separate 'mean time to happen' for the red marbles. But you do not want to change the overall chance of any marble to appear; the combined probability of a red or green marble appearing must be the same as the previous probability of a green one appearing.

Aim at, say, a fifteen percent probability of a red marble appearing instead of a green one, with the current 'mean time to happen' being one thousand seconds. What 'mean times to happen' would you have to assign to both green and red?

Edited July 22, 20187 yr by Mouthwash

The best approach would be to use A Poisson distribution (that seems to be what you are describing.  Let t be time with unit one thousand seconds.  Then the probability density for a mare to appears is e^{-t} To split it up, the density function for the red is .15e^{-.15t} while for the green is .85e^{-.85t}.  The mean times for the red an green respectively are 1000/.15=6666.66... and 1000/.85=1176.47 seconds.

Edited July 23, 20187 yr by mathematic
Latex?

The values I have are 1000, 2000, 800 and 1600. To do 2000 I just need to double your results to be 2352.94 for green and whatever the twice 6666.6 recurring is for red, right?

(Is it possible to double a recurring number? It's shockingly hard to find that out.)

But I'd like to know the values for 800 and 1600. Not really up to doing this myself, sorry.

 

8 hours ago, Mouthwash said:

The values I have are 1000, 2000, 800 and 1600. To do 2000 I just need to double your results to be 2352.94 for green and whatever the twice 6666.6 recurring is for red, right?

(Is it possible to double a recurring number? It's shockingly hard to find that out.)

Convert that part to a fraction, then multiply.

ex. 2 * 1/3

Edited July 23, 20187 yr by Endy0816