What is the probability of two independent events occurring simultaneously?

[oneamp]

Hello. I have two probability density functions for two events. I would like to find the probability that they both will occur at the same time. It is simply multiplying the results of the two integrals over the time, correct?

Thank you

 

[mathman]

Hello. I have two probability density functions for two events. I would like to find the probability that they both will occur at the same time. It is simply multiplying the results of the two integrals over the time, correct?

Thank you

No. Two events can occur at the same time only if they have discrete distributions. When both have continuous distributions (and are independent) the probability of happening at the same time is 0.

 

[oneamp]

Yes true :) How can I calculate the probability that between some points 'a' and 'b' in time, two events with these PDFs will occur? For example, if one PDF describes the probability that a light will be orange, and another PDF describes probability for a green light, and I want to know the chances that there will be an orange and a green light illuminated "at the same time" between times 'a' and 'b'?

 

[Office_Shredder]

oneamp, if the two events are independent then you can simply multiply the probabilities of each even happening. If they have some dependence between each other then you need to know exactly what that dependence is - you have to have a pdf p(x,y) which is called the joint distribution between the two variables.

 

[oneamp]

Thank you very much

 

[jbunniii]

No. Two events can occur at the same time only if they have discrete distributions. When both have continuous distributions (and are independent) the probability of happening at the same time is 0.

That's not true at all. Suppose $X$ and $Y$ are independent normally distributed random variables. Let $A$ be the event that $X > 0$ and $B$ be the event that $Y > 0$. Clearly the probability of $A \cap B$ is nonzero.

 

[mathman]

That's not true at all. Suppose $X$ and $Y$ are independent normally distributed random variables. Let $A$ be the event that $X > 0$ and $B$ be the event that $Y > 0$. Clearly the probability of $A \cap B$ is nonzero.

You misunderstood the point of the original question. He was asking about something like the probability that X=Y when both have continuous distributions.