Question

what's the chance of seeing 4 double-digits on a mobile phone's digital clock during a 17hr waking period between 07:07 & 23:23hrs; eg. 07:07, 16:16, 23:23.

Answer 1

To determine the chance of seeing double-digit times on a digital clock within the 17-hour period from 07:07 to 23:23, we start by identifying all the possible times that meet the double-digit criteria during this window:

### Definitions:
- **Double-digit time**: Times where the hour and minute each are at least 10 (i.e., for hours 10:00-23:59, but with the additional restriction of being within the bounds of 07:07 and 23:23).

### Double-digit Analysis:
In a 24-hour format, the times with double-digit hours that we can consider are:
- 10:00 to 10:59
- 11:00 to 11:59
- 12:00 to 12:59
- 13:00 to 13:59
- 14:00 to 14:59
- 15:00 to 15:59
- 16:00 to 16:59
- 17:00 to 17:59
- 18:00 to 18:59
- 19:00 to 19:59
- 20:00 to 20:59
- 21:00 to 21:59
- 22:00 to 22:59
- 23:00 to 23:23 (restricting minutes to the first 23 minutes)

### Range of Double-Digit Times:
- **10:00 to 10:59**: 60 minutes
- **11:00 to 11:59**: 60 minutes
- **12:00 to 12:59**: 60 minutes
- **13:00 to 13:59**: 60 minutes
- **14:00 to 14:59**: 60 minutes
- **15:00 to 15:59**: 60 minutes
- **16:00 to 16:59**: 60 minutes
- **17:00 to 17:59**: 60 minutes
- **18:00 to 18:59**: 60 minutes
- **19:00 to 19:59**: 60 minutes
- **20:00 to 20:59**: 60 minutes
- **21:00 to 21:59**: 60 minutes
- **22:00 to 22:59**: 60 minutes
- **23:00 to 23:23**: 24 minutes

### Total Double-Digit Minutes:
Calculating the total minutes for the above:
- From 10:00 to 22:59: 13 hours × 60 minutes = 780 minutes
- From 23:00 to 23:23: 24 minutes

Total double-digit minutes = 780 + 24 = **804 minutes**.

### Total Waking Minutes:
The total time from 07:07 to 23:23 is:
- **23:23 - 07:07 = 16 hours and 16 minutes**.
- 16 hours = 960 minutes + 16 minutes = 976 minutes.

### Probability:
If we want to find the probability of seeing a double-digit time in that window, we now set up the expression.
- Total possible moments to see the clock = 976 minutes.
- Favorable moments to see a double-digit time = 804 minutes.

Calculating the probability:
\[ \text{Probability} = \frac{\text{Favorable Outcomes}}{\text{Possible Outcomes}} = \frac{804}{976} \approx 0.821, \text{ or } 82.1\% \]

### Conclusion:
Thus, the chance of seeing double-digit times on a mobile phone's digital clock during the specified period is approximately **82.1%**.