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## Time's A-Wasting

Probability puzzles require you to weigh all the possibilities and pick the most likely outcome.

 Puzzle ID: #40540 Fun: (2.27) Difficulty: (2.03) Category: Probability Submitted By: kratsg

A digital clock tells the time in hours and minutes. During what fraction of the day does it display at least one '1'?

720/1440 or 1/2

Here's the systematic approach to getting the answer:

Convert all the times of the day into simple numbers: 1:00 = 100, etc...

From 100-159, there are 60 times that there is at least one '1'.

This is the same for the ranges of 1000-1059, 1100-1159, and 1200-1259.

From 200-259, there are 15 times that there is at least one '1'

This is the same for ranges of 300-359, 400-459, 500-559, 600-659, 700-759, 800-859, and 900-959.

That means 8 ranges have 15 times, 4 ranges have 60 times. We'll double this to convert to a 24-hour day:

2[(15)(8) + (60)(4)] = 720 total times that there is at least one '1'.

To figure out the total number of times that any number is shown at least once, we know there are 60 times between any range listed above, so we simply multiply by 24 to get the total number of times in 24 hours.

(24)(60) = 1440 total times.

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