Am I just lucky?

The purpose of the problem is to show that many coincidences are not that rare when you take a close look at them.

First, let's count how many "singular" times we have in the period. Take the hour 15, for example. There are 3 singular times: 15:00, 15:15 and 15:51. Let's suppose this also happens for every hour in the 14-hour period (3 x 14 = 42). There are exceptions we have to deduct, though – hours that have only 2 of these 3 possible configurations: 11 and 22 (permutating A and B makes no changes) and 16 through 19 (where $B>5$ , there is 19:00 and 19:19 but not 19:91). The total of singular times is then $3\times 14-6=\textbf{36}$ .

One way of estimating the desired probability is to think of the problem as equivalent to picking 30 different times at random from a set of 840. The probability of having at least one singular time among the 30 picked is the complementary probability of not having any:

$p=1-\dfrac{{840-36}\choose 30}{840\choose 30}\approx \boxed{0.74}$

Another way is to see that there is a total of ${36}$ favorable cases out of ${840}$ possible ones. The probability of encountering any of those times when we check the phone once is $P=\frac{36}{840}=\frac{3}{70}$ . The probability of encountering at least one of such times in 30 checks (supposing times can be picked more than once) is:

$p=1-(1-P)^{30} \approx \boxed{0.73}$

This probability is surprisingly high indeed! The odds were clearly in my favor after all.

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• Notice the second value is very similar to the first. Allowing the times to be picked more than once to make the events independent does not make much of a difference since $30 \ll 840$ .