Double Digits
Probability puzzles require you to weigh all the possibilities and pick the most likely outcome.
As an inveterate clock watcher, James glances at his digital watch at random from time to time throughout the day. What is the probability that he will see all the digits reading the same (for example, 4:44) during any single glance? (Assume a typical day, not a day on which daylight savings time changes or anything like that. The watch is set to a 12 hour U.S. format, not a 24 hour European or military format.)
HintHow many times does the given occurrence happen in a day? How long do these occurrences last? How long is this compared to a day?
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Answer
There are 12 instances in each 24 hour period when a digital clock or watch will show all identical digits. (1:11,2:22,3:33,4:44,5:55 and 11:11, each times two = 12) for exactly a minute each instance. Since there are 1440 minutes (24x60) in a 24 hour period, the probability of James seeing all the same digits in any 24 hour period is 12/1440 = 0.0083 or slightly less than one chance out of a hundred.
Practically speaking, however, since most of us are asleep from 11:11 pm through 5:55 am each NIGHT, James PROBABLY did not glance at his watch during the night hours. Suppose that James sleeps 8 hours every day, the most correct answer would then be 6/960, or about 1 chance in 160. If that (or 0.00625) was your answer, give yourself extra credit for not viewing this as strictly a math problem.
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Comments
pizzahead
Oct 17, 2001
 Find me a watch that shows 00 instead of 120. It happens 12 times, not 14. It's .0083, not .0097. 
Maneesh
Nov 16, 2001
 If you are considering sleeping time then also probability will not change.If favourably chances are reduced then dividing factor will also get reduced. 
abrakadabra_12
Dec 05, 2001
 I AGREE wit pizzahead! i too wanna see the watch that gives you the tyme 00?
~$abrakadabra$~ 
(user deleted)
Feb 10, 2002
 Agreeing with the 12 occurances, I must disagree with the calculation of probability for Jim, unless this inveterate "watcher" looks at the time "randomly" once during every minute cycle, in which case I might label him as OCD (obsessivecompulsive disorder). Anyway, there is a 12 in 1440 chance, approx. 0.83% (obviously) of the time occurances, but Jim's probability can only be based on the number of actual times he looks at his watch during the day, so if he looks at the reading 30 times per day, there would be a 25% chance (0.83*30)of him seeing identical digits. 
bighippo4
Apr 04, 2002
 You said that the time is American time, there IS no "00" in standard time, therefore the correct answer, assuming the person does not sleep is 1/120. 
pusandave
Apr 05, 2002
 I assumed James had insomnia, but I forgot the 11:11 time. Good one, well done! 
chamber44
May 17, 2002
 lamentor, read the question again. It clearly stated "what are the chances...DURING A SINGLE GLANCE". That's it. Also, I agree that there is no 00 on a digital watch. 
opqpop
Sep 29, 2010
 Good problem. It taught me to just find number of minutes in 12 hours to get total # of possible times instead of count them w/ multiplication. 
pauliD
Mar 04, 2016
 Don't know if anyone is still reading this. How's about taking this teaser one step further. What is the probability of looking at your watch and having the minutes and seconds match. and if one had the ability to see matches more frequently than predictable, what would that indicate.? 
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