Help with intuition on the exponential distribution

Question

My book says an exponential random variable is good for modelling the waiting time until an event occurs. However, I'm having a bit of difficulty understanding why that should be the case, given the shape of the distribution. For instance, let $t\text{~}expo$ denote the time (in minutes) until a baby is born in a hospital where, on average, a baby is born every $15$ minutes. So the PDF for $t$ is given by

$$f(t)=\frac{1}{15}e^{-\frac{1}{15}t}\text{,}$$

right? However, this implies, for instance, that P$(1\le{}t\le{3})$$>$P$(14\le{}t\le{16})$, which goes strongly against intuition: if the average rate of birth is $15$ minutes, shouldn't we expect a birth within $14$ to $16$ minutes from now to have a higher probability than one within just $1$ to $3$ minutes? I know I'm missing out something quite fundamental (and possibly silly) here. Can someone please tell me what it is? Thanks very much in advance.

Answer 1

Your intuition is based on the comparison of two probabilities as shown below.

If you take the mechanical example and consider the expectation to be the center of mass then the paradox will disappear. You simply take into account that the location of the center of mass depends not only from the weight of the individual pieces but also from the location of those pieces.

Now, your question, in the language of mechanics:

"How come that near the center of mass the mass is less than somewhere else?"

Answer 2

The probability that the next birth will be between $1$ and $3$ minutes from now is greater than that it will be between $14$ and $16$ minutes from now. The probability that there will be a birth in any $2$ minute interval is independent of the interval.

Your intuition about repeated coin flips might help. The probability that there will be a head in tosses $5$ or $6$ is the same as the probability of a head in tosses $15$ or $16$ but the probability of the first head is clearly greater for the earlier interval than the later one. That's true even if the coin is unfair and comes up heads with probability only $1/15$.

What you're missing is in fact fundamental, but missing it isn't silly. The exponential distribution is really memoryless. I think you imagine that the rate of $1/15$ means something like "trains arrive every fifteen minutes on average". Well coins come up head half the time on average, but that's not because heads and tails are somehow scheduled to alternate, although on average they do. Wrapping your head around real randomness is hard.