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probability of seeing 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.

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To find the probability of seeing double-digits on a mobile phone's digital clock during a 17-hour waking period from 07:07 to 23:23, we first need to identify the occurrences of double-digit times within that range.

Double-digit times are formed when both the hour and minute digits are the same, e.g., 07:07, 08:08, 09:09, 10:10, 11:11, 12:12, 13:13, 14:14, 15:15, 16:16, 17:17, 18:18, 19:19, 20:20, 21:21, 22:22, and also 23:23.

List of Double-Digit Times Within the Range:

  1. 07:07
  2. 08:08
  3. 09:09
  4. 10:10
  5. 11:11
  6. 12:12
  7. 13:13
  8. 14:14
  9. 15:15
  10. 16:16
  11. 17:17
  12. 18:18
  13. 19:19
  14. 20:20
  15. 21:21
  16. 22:22
  17. 23:23

Total Count of Double-Digit Times

In the specified range (07:07 to 23:23), we have:

  • From 07:07 to 07:07: 1 occurrence (07:07)
  • From 08:08 to 08:08: 1 occurrence (08:08)
  • From 09:09 to 09:09: 1 occurrence (09:09)
  • From 10:10 to 10:10: 1 occurrence (10:10)
  • From 11:11 to 11:11: 1 occurrence (11:11)
  • From 12:12 to 12:12: 1 occurrence (12:12)
  • From 13:13 to 13:13: 1 occurrence (13:13)
  • From 14:14 to 14:14: 1 occurrence (14:14)
  • From 15:15 to 15:15: 1 occurrence (15:15)
  • From 16:16 to 16:16: 1 occurrence (16:16)
  • From 17:17 to 17:17: 1 occurrence (17:17)
  • From 18:18 to 18:18: 1 occurrence (18:18)
  • From 19:19 to 19:19: 1 occurrence (19:19)
  • From 20:20 to 20:20: 1 occurrence (20:20)
  • From 21:21 to 21:21: 1 occurrence (21:21)
  • From 22:22 to 22:22: 1 occurrence (22:22)
  • From 23:23 to 23:23: 1 occurrence (23:23)

This gives a total of 17 double-digit occurrences within the specified time range.

Total Number of Minutes in 17 Hours

Next, we calculate the total number of minutes in the waking period from 07:07 to 23:23:

  • The starting time (07:07) to the end time (23:23) is 16 hours and 16 minutes.
  • The total time in minutes is \( (16 \times 60) + 16 = 960 + 16 = 976 \) minutes.

Probability Calculation

Now, we can calculate the probability of seeing a double-digit time.

\[ \text{Probability} = \frac{\text{Number of Double-Digit Occurrences}}{\text{Total Minutes in Waking Period}} \]

\[ \text{Probability} = \frac{17}{976} \]

Final Result

The probability of seeing double-digit times on a digital clock during the specified 17-hour waking period from 07:07 to 23:23 is approximately:

\[ \text{Probability} \approx 0.0174 \text{ (or } 1.74% \text{)} \]