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.
- If you stare at the clock for 6 seconds beginning at a random starting time, what is the probability you will observe it changing?
- 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?
- 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?
- 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?
(*): If you can't get 50% exactly, get as close as you can.