While bored in class one day I started to come up with a problem that I kept making more difficult as I solved each step. The overall setup goes like this. Picture an alarm clock or a scoreboard clock. There are 5 sections of the displayed time. 4 places are for where numbers go and the middle, and last place, is for the colon. Each number slot contains 7 elements/segments that allows it to create any of the 10 digits. I also divided the colon into 2 pieces. All 30 elements/segments work independently of each other. For each problem I don't consider anything without a lit colon to be a normal time. Lastly by displayed I mean lit and by not displayed I mean it's turned off. To give you a sense of how to attack problem 4 I'll show you each problem that led to me to it. I don't want an answer regarding any problem other than 4 because I've already solved the other ones. This forum seems like a group of smart people so I came here with my challenge first. 1st problem. Given that the probability a segment being lit or not lit is fair what is the probability that a normal time is displayed. By normal time I mean a mod 12 time, not military time, so anywhere from 1 o'clock to 12:59. This didn't take me too long to do 2nd problem. Now consider that a segment has, instead of 2 like the last problem, 3 settings. The settings are off, on-low, on-high. So a segment that is on-low will be dimmer than one on-high. The probability is still fair for all 3. What is the probability that a normal time is displayed? What is the probability that a normal time is displayed and all the displayed segments are on the same setting? This took me a bit longer but it's really not that tough to get 3rd problem. Now consider that a segment has, instead of 2 or 3 like the last problems, n settings. The settings are off, on-1, on-2,...,on-(n-1) where on-k is dimmer than on-j if k<j. What is the probability that a normal time displays? What is the probability that a normal time displays and each displayed segment is lit on the same setting? This one I had to finish outside of class 4th problem. We will use the same setup as problem 3. Let's say that if a segment is turned off that its value is 0 and if it's on-1 its value is 1 and if it's on-i its value is i. What is the probability that a normal time displays AND its total value is λ for any integer valued λ. I've done some work on this one but overall it seems beyond me. If there's anything missing or wrong please tell me and I will fix it. Thanks so much and good luck.