How could you distinguish it being less than ⅕ and it being less than or equal to ⅕. Certainly they're different events, in fact the first one is contained in the second one, so how can you distinguish these 2 events
Great question! The answer is that these two events have the same probability, since the probability of the point being chosen exactly at 1/5 is zero (since there are a continuum of choices). For example, if I throw a dart at the interval [0,1], the probability it exactly lands at 0.5 is zero. It might land really close to 0.5, like within the interval [4.99999,5.000001], with nonzero probability, but the probability that it exactly, perfectly lands on 0.5 is zero.
I guess what I don't like about this is that there is, after all, a chance that the dart would hit exactly 0.5. So I'm not realy sure how to balance this intuition vs the math meaning of it
@@gsscala i think what he's trying to say is that the odds of landing on any specific point are so low (because there is an infinite number of points), the odds are practically 0
@@gsscala "I'm not realy sure how to balance this intuition vs the math meaning of it" You balance it by accepting the following: Impossibility implies a probability of zero, but a probability of zero does not imply an impossibility. You are trying to equate the two concepts, but the implication only holds in one direction. Here is a geometric perspective. A probability distribution is defined by a function f where the probability of drawing a value in some span from a to b is the area under the curve from a to b. Impossibility: If values in the span a to b are impossible then f has a height of zero over that span, which implies zero area, which implies zero probability. Single possible value: Here, we would have a = b, so our width is zero, implying the area is zero, implying the probability is zero. However, the value a is still possible. (This supposes f is finite; see the Dirac delta function if you want to get weirder.)
How could you distinguish it being less than ⅕ and it being less than or equal to ⅕. Certainly they're different events, in fact the first one is contained in the second one, so how can you distinguish these 2 events
Great question! The answer is that these two events have the same probability, since the probability of the point being chosen exactly at 1/5 is zero (since there are a continuum of choices). For example, if I throw a dart at the interval [0,1], the probability it exactly lands at 0.5 is zero. It might land really close to 0.5, like within the interval [4.99999,5.000001], with nonzero probability, but the probability that it exactly, perfectly lands on 0.5 is zero.
I guess what I don't like about this is that there is, after all, a chance that the dart would hit exactly 0.5. So I'm not realy sure how to balance this intuition vs the math meaning of it
@@gsscala i think what he's trying to say is that the odds of landing on any specific point are so low (because there is an infinite number of points), the odds are practically 0
@@gsscala "I'm not realy sure how to balance this intuition vs the math meaning of it"
You balance it by accepting the following: Impossibility implies a probability of zero, but a probability of zero does not imply an impossibility.
You are trying to equate the two concepts, but the implication only holds in one direction.
Here is a geometric perspective. A probability distribution is defined by a function f where the probability of drawing a value in some span from a to b is the area under the curve from a to b.
Impossibility: If values in the span a to b are impossible then f has a height of zero over that span, which implies zero area, which implies zero probability.
Single possible value: Here, we would have a = b, so our width is zero, implying the area is zero, implying the probability is zero. However, the value a is still possible. (This supposes f is finite; see the Dirac delta function if you want to get weirder.)
It actually is, your reasoning can be applied only to discrete intervals, while a [0,1] is continous, so it can be divided in intervals and so on