Very interesting read! You made the comment, "So the probability of obtaining a specific value is always zero (or infinitesimally small)." You then follow that up by saying it's not feasible to calculate the probability of any specific value because it's such a small number, which is why it's better to use probability density around each specific value instead. (I hope I am interpreting this correctly)
I thought about this, and this is 100% correct. However, consider this: every occurrence on the graph is technically a specific value in itself, no different from 3 being a specific value. In that argument, arriving at exactly 3 possesses a similar probability to any of the outcomes that are closely distributed around the value of 3. Yet, like you said, the likelihood of achieving any one of those 'specific values' is pretty much impossible, making any outcome that occurs when you finally arrive at point B, both a miracle, and not a miracle at the same time, because your specific arrival time is almost impossible odds.
But then you have the theoretical argument (since you brought in the infinite nature of the math) that since there are an infinite number of values that lie between the 1 and 5 minute arrival time, the term "specific value" is only a limitation of our current math, because each value in itself is infinitely more specific. For example, the value of 3 is just a round number, and the specific value could be 3.00001. But that's only viewing the value through a lens of 5 decimal places, and you could keep breaking the number down infinitely. This backs your argument of 0% probability because in the world of infinity, there is really no such thing as a specific value. We create those values based on our own mathematical limitations. (Which I believe is precisely what you are saying.
Very very good.
Thanks for appreciating, Jean :)
In this case, the most important is to understand that probability in continuous random variables is area under curve of the distribution
Absolutely, Walter :)
Thanks for highlighting that.
Very interesting read! You made the comment, "So the probability of obtaining a specific value is always zero (or infinitesimally small)." You then follow that up by saying it's not feasible to calculate the probability of any specific value because it's such a small number, which is why it's better to use probability density around each specific value instead. (I hope I am interpreting this correctly)
I thought about this, and this is 100% correct. However, consider this: every occurrence on the graph is technically a specific value in itself, no different from 3 being a specific value. In that argument, arriving at exactly 3 possesses a similar probability to any of the outcomes that are closely distributed around the value of 3. Yet, like you said, the likelihood of achieving any one of those 'specific values' is pretty much impossible, making any outcome that occurs when you finally arrive at point B, both a miracle, and not a miracle at the same time, because your specific arrival time is almost impossible odds.
But then you have the theoretical argument (since you brought in the infinite nature of the math) that since there are an infinite number of values that lie between the 1 and 5 minute arrival time, the term "specific value" is only a limitation of our current math, because each value in itself is infinitely more specific. For example, the value of 3 is just a round number, and the specific value could be 3.00001. But that's only viewing the value through a lens of 5 decimal places, and you could keep breaking the number down infinitely. This backs your argument of 0% probability because in the world of infinity, there is really no such thing as a specific value. We create those values based on our own mathematical limitations. (Which I believe is precisely what you are saying.