How does one formulate continuous probabilities/pdfs?

[random_soldier1337]

Discrete examples are easy enough. Toss a coin, 1/2, toss a die, 1/6.

Continuous examples, Probability of a nucleus decaying during observation, 1-exp(-λt), Probability of a neutron moves x without interaction, exp(-Σx), where Σ can be assumed to be the inverse of the mean free path i.e. the distance a neutron travels without interaction on average.

My point is that I don't really have an idea as to how these continuous probabilities are derived. Any assistance?

[studiot]

15 minutes ago, random_soldier1337 said:

My point is that I don't really have an idea as to how these continuous probabilities are derived. Any assistance?

What does the probability tell you?

Why is the probability of getting heads in a coin flip one half and what does that mean?

Once you have this clear you can move on to probabilities of continuous distributions.

Edited October 6, 2018 by studiot

[random_soldier1337]

Well in the case of the coin it tells me that there are 2 random events that can happen when I flip a coin. The chance that one or the other takes place is 1/2 under the assumption both sides are affected by the same unbiased factors.

Now, taking for example the probability of a nucleus decaying, I know that the number of nuclei in a sample N = N_oexp(-λt), where N_o is the initial amount. Looking at the probability of a nucleus decaying another way, I could put it as 1-N/N_o. I'm not really sure I understand. 1- the percentage of nuclei remaining at a given time gives me the probability?

EDIT: Or do you mean to say that they are derived by observation and large sample sizes of a phenomenon?

Edited October 6, 2018 by random_soldier1337