Adding a Battery to a Clock and the Probability of Having the Time Be Exactly Correct OK, this is probably basic for probability gurus; but, I just grabbed a cheap wall clock from Target, brought it home, and went to put it on the wall--having to add one AA battery before hanging it, of course. 
Well, when I went to change the time, I noticed that it was exactly 2:03PM on my cell phone and exactly 2:03PM on the wall clock. Moreover, when the phone switched to 2:04, the second hand was at 2:03:59 . . .!
Shocked so much by how crazy that was, I had to dig through the trash to find the packaging to make sure it wasn't some kind of satellite-linked clock--sure enough, just a $12 P.O.S. clock from Target.
Anyway, my question is, seriously, "What's the probability of something like that happening?" Or, put more simply, what's the probability that when you add a battery to a clock, it'll be set to the same time that it actually IS? (Keep in mind this is a 12 hour wall clock; so, there's no way to say for certain that it was AM or PM just looking at the clock).
 A: The probability of being the right hour is $\frac 1{12}$ of being the right minute $\frac 1{60}$ and of being the right second $\frac 1{60}$
So the total probability of switching on the clock at the exact second of another timepiece (disregarding AM/PM) would be $\frac 1{60 * 60 * 12} = \frac 1{43200}$
But, this is assuming that the probability density of each is randomly distributed. That is a pretty big assumption. 
A: The question is "What's the probability of something like that happening?" and thus should be answered under the appropriate scope of "something like that".
By a conservative estimate, each of the $7$ billion people on Earth on average hangs up one wall clock in their life, which lasts approximately $3$ billion seconds. Thus, roughly $2$ people are hanging up a wall clock per second. If a day has $n$ seconds, that makes $2n$ people per day.
As derived in the other answers, the probability of accidentally starting a $12$-hour wall clock exactly set to the right second is $2/n$. The probability that none of the $2n$ people start the clock exactly on the right second is thus
$$
\left(1-\frac2n\right)^{2n}\approx\mathrm e^{-4}\approx2\%\;.
$$
Already for a single day, this is quite unlikely. Over the course of, say, a year, it becomes extremely unlikely. The expected number of people who set their wall clocks exactly right over the course of a year is roughly
$$
365\cdot2n\cdot\frac2n=1460\;.
$$
math.SE has been around for five years, so over the course of its existence roughly $7300$ people have set their wall clocks exactly to the right second by accident. It was only a matter of time (pun intended) until one of them would find their way here and ask this question.
A: Since there are $12 \times 60 \times 60 = 43200$ seconds per 12 hours the probability it was set to the very same second is $1/43200\approx 0.0023$ %. If it was exact to 1/10th second then the probability drops by 1/10.
