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This is a relatively straightforward little puzzle, but the answers are somewhat unintuitive or can prompt a new perspective. It might be an interesting little puzzle for a probability teaching example or job interview question.
I hope it helps get you thinking, in an useful way!

For the purpose of this puzzle, you have a digital clock that displays hours and minutes.
The clock [display] changes instantaneously after 60 seconds, every 60 seconds (which you can approximate to 59.9999999999... if that helps below). For periods relevant to this puzzle, the clock is guaranteed to work perfectly and without ceasing.

  1. If you stare at the clock for 6 seconds beginning at a random starting time, what is the probability you will observe it changing?
  2. If you do what's described in #1 ten times independently of each other, what is the probability you will observe the clock changing? (Guidance question under spoiler tag).

    Should this be higher than, lower than, or the same as the answer to #1?

  3. If you wanted the probability of observing a change to be 50%*, how many times would you have to do what's described in #1?
  4. If the ten times described in #2 are sequential, so that you're watching for one continuous period, then what is the probability you will observe the clock changing?

Have fun!

(*): If you can't get 50% exactly, get as close as you can.

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closed as off-topic by ffao, JonMark Perry, noedne, Rubio May 21 '18 at 16:32

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "This question is off-topic as it appears to be a mathematics problem, as opposed to a mathematical puzzle. For more info, see "Are math-textbook-style problems on topic?" on meta." – ffao, JonMark Perry, noedne, Rubio
If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ It's true that, often, statistics is unintuitive. But that doesn't make it a puzzle. $\endgroup$ – Rubio May 21 '18 at 16:34
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    $\begingroup$ @Rubio It seemed on topic based on this question, this question, this question, this question, this question, and questions related to those, which are straightforward, usually (but not always) unintuitive, applications of statistics. Should all of these be moved/closed? $\endgroup$ – WBT May 22 '18 at 23:08
  • $\begingroup$ The questions you linked all look to be puzzles, not straightforward application of expected mechanics that would be well known to anyone reasonably familiar with the topic. See in particular this question and its answer for an overview of Math vs Puzzling, and follow the links therein for more background if desired. If the unintuitive element in a question is only unintuitive to someone with a lack of familiarity with the subject, that’s not the kind of puzzling element we’re looking for. Not everything someone finds puzzling is a puzzle. $\endgroup$ – Rubio May 23 '18 at 0:46
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1.

You will observe the clock change time if you look at it any time from (instantaneously) after 6 seconds until right before the clock would normally change (aka xx:xx:00), so the probability is $\frac{6}{60}=\frac{1}{10}.$

2.

The probability that you don't see the clock change in one glance is $1-\frac{1}{10}=\frac{9}{10}.$ So the probability that you don't observe the clock change in any of those 10 instances is $(\frac{9}{10})^{10}$ and so the probability that you do see the clock change is $\boxed{1-\left(\frac{9}{10}\right)^{10}}$ which is around $0.651.$

3.

We're basically trying to solve $1-(0.9)^n=0.5$, so $\boxed{n=7}$ (actually, n comes out to $\log_{0.9}0.5,$ which is around $6.579$).

4.

This is basically equivalent to watching the clock for a minute. That means that no matter what, at some point in the interval you'll observe the clock changing (unless you want to count the period from $0.000\dots 1$ seconds past the minute to $59.999\dots$ seconds past the minute as a valid interval), so the probability is $\boxed{1}.$

This definitely feels more like math than puzzles, tbh.

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  • $\begingroup$ I think it's the type of puzzle that can introduce a math lesson and promote mathematical thinking/curiosity. $\endgroup$ – WBT May 17 '18 at 15:41
  • $\begingroup$ I'm suggesting that this would probably fit better under math.stackexchange.com, as math puzzles tend to be frowned upon in this stackexchange, but I guess it's fine for now unless moved. $\endgroup$ – pie314271 May 17 '18 at 15:42
  • $\begingroup$ (Also, thanks to whoever fixed the spoiler in #1) $\endgroup$ – pie314271 May 17 '18 at 15:58
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1.

1 in 10, so 10%

2.

This depends on whether you can time the intervals or not. If you take 10 consecutive intervals of 6 seconds (or choose them $60 \cdot n$ seconds apart, where $n$ is a natural number), you're guaranteed to see a change, so it's 100%. If the intervals are random and independent, the chance that you don't see a change is $(9/10)^{10} \approx 0.35$, so the chance that you do see at least one change is 1 minus that, so $\approx 0.65$.

3.

Assuming the intervals are chosen independently, the solution is $\frac{\log(50%)}{\log(9/10)} \approx 6.58$. So 6 is too few times, and 7 too many, but the answer for 7 (52.2%) is closer than the one for 6 (46.9%).

4.

See the answer to 2.

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