Why should the time between events be modelled by an exponential probability function?

There was a question on a past exam which went something like this.

106 points are scored in a basketball game

the game lasts for 80 minutes

find the probability density function of the time between two points

In the question there was nothing about exponential distributions, and to me it made sense to use a constant rectangular pdf because the probability of scoring a point at a time should be independent of any other time. I know there are other factors in basketball but ignore that.

The answer in the solutions was an exponential pdf, and because there was nothing in the question to suggest that the solution should have been an exponential pdf, im assuming theres a good reason why its exponential.

If this is the case, can someone please try explain to me why "time between events" is represented by an exponential pdf. Thanks.

Why should the time between events be modelled by an exponential probability function?

Assume events are happening according to a Poisson process with rate λ. Let t be any point in time. Let T be the waiting time until the first event occurs after time t. Then T has the exponential distribution with parameter 1/λ.

Proof:

Let x > 0; P(T > x) = P(no event in time interval of length x)

use the Poisson distribution

= exp(-λx) * (λx)^0 / 0! = exp(-λx)

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Why should the time between events be modelled by an exponential probability function?

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