I am a bit confused on the Exponential prob density function. I am clear on a simple problem involving a poisson arrival mechanism and how to determine the probability of an arrival (or event) within a given timeframe. However, I got tripped up when making a small variation on the problem. Let me illustrate with this example.

First, the simple problem:

A tennis player is practicing with a tennis ball machine. The machine is set to throw the yellow balls randomly at a mean delivery rate of 1 every 6 seconds. What is the probability of the player receiving a ball within 2 seconds?

Easy: P(T <= 2) = 1 - exp(-2/6) =~ 28.34 %

Now, the harder problem:

Someone rolls onto the court a second machine that also delivers balls at a rate of 1 every 6 seconds. The second machine delivers white balls.

I am certain that the probability of the player receiving a ball of any color within 2 seconds is:

P(T <= 2) = 1 - exp(-2*2/6) =~ 48.66 %

However, I am stuck with the observation that the probability of receiving a white ball is:

Pw(T <= 2) = 1 - exp(-2/6) =~ 28.34 %

and a yellow ball is:

Py(T <= 2) = 1 - exp(-2/6) =~ 28.34 %

which add up to more than 48.66%.

I know that there is an explanation of this and a way to harmonize the equations. I know that the equations are multiplicative and not additive. i.e. the probability is:

P(T <= 2) = 1 - (1-Py)*(1-Pw) = 1 - exp(-2/6)^2 = 1 - exp(-2/6-2/6)

=~ 48.66 %

What I cannot get my head around is the actual explanation of this in statistical terms. I suspect that the one machine model and two machine model are actually different and that I should not even be using Exponential PDF for the two machine model. What leads me to this conclusion is that one machine cannot throw two balls simultaneously.

But, then again, does the exponential PDF allow 2 balls to be thrown simultaneously by one machine? Actually, I think it does. In that case, the exponential pdf is wrong in both cases.

Do you see my dilemma?

Thanks and Brgds,

Mark