Probability of a watched clock changing.

[BonJoviJones]

This is a generic probability question I came up with while watching the digital clock in my car waiting for it to warm up:

You look at a digital clock (that doesn't display seconds) for 1 second. What is the chance that it changes in that second? Assuming it doesn't change and you continue to observe it, what is the chance it changes in the following second... and the following second... etc etc.

I think it's pretty clear that for the first second the chance is 1 in 60. Also, while observing on the 60th second, it _must_ change. This implies P(change) is 1/n, where n is (60 - the number of seconds observed).

Can anyone answer for sure?

Thanks.

[daryn]

a more interesting question i think is what is the probability of a watched pot boiling

[MrBlini]

The probability density of clock changes for the next 60 seconds from a point in time selected at random with equal probability of being at each point in the minute interval is uniform:
p(t) = 1/60, 0<=t<=60
The cumulative density function is
P(T<=t) = t/60, 0<=t<=60
This is the probability of observing the change in t seconds.

The conditional probability of observing a change in the next second given that no change has occurred yet in the last t seconds is
P(t<T<=min(t+1,60) | T>t) = P(t<T<=min(t+1,60) and T>t) / P(T>t)
= P(t<T<min(t+1,60)) / P(T>t)
= (min(1,60-t)/60) / (1-P(T<=t))
= (min(1,60-t)/60) / (1-t/60)
= (min(1,n)/60) / (n/60)
P(Change) = min(1,n)/n, 0<=n<=60

The need for the min function arises from the fact that the remaining interval is less than one full second if less than one second remains in the observed minute.

[SossMan]

If I stare at the hottie across the bar long enough, will she ask me to her place for some "systematic number crunchin'"? [img]/images/graemlins/heart.gif[/img]