A friend of mine posed this problem and we have had a disagreement on the answer.
- There is a 90% chance that some event will happen in the next year.
- There is a 95% chance that the event will happen eventually.
- What is the probability that it happens after this year, if it does not happen in the next year?
Hover over here for a description of our answers and arguments:
I told my friend that it is 95%. My reasoning is that there is a 95% chance that it happens from now to infinity. The probability that it happens from now to infinity minus one year is still 95%. In one year, when it hasn't happened, events that didn't happen don't affect your "eventual" odds.
My friend on the other hand, believes it to be 50%. He had two arguments. First, he said if you flip a coin 4 times, the chance that you get heads the first time is 50%, but the chance that you get heads eventually is 93.75% (15/16). If you don't get it the first time, what is the chance that you get it after the first time? 87.5% (7/8), in other words, it has decreased.
To this I responded that the "eventually" is based on a finite set of events, each with their own probability and after the first event has passed, you have fewer chances. In his problem, on the other hand, it is a summary of the probability of unknown events broken in two time periods.
He continued to insist that it is the same because you have two time periods with known odds. The 95% is the combination of two probabilities, 90% and 50%. Eliminating the first 90%, the probability of the remaining period is 50%.
I agreed that's an accurate way to look at it from now, but I don't think that's accurate once you know it did not happen in that time period.