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Two witches make a nightly visit to an all-night coffee shop. Each arrives at a random time between 0:00 and 1:00. Each one of them stays for exactly 30 minutes. On any one given night, what is the probability that the witches will meet at the coffee shop?

So two links below have provided conceptually different answers, of course, the time where they stay in each circumstance, which are 15 and 30 minutes, is of second importance.

http://brainstellar.com/puzzles/30 https://puzzlefry.com/puzzles/witches-at-a-coffee-shop/

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    $\begingroup$ Wouldn't this be a more stats kind of question? $\endgroup$ – Abbas Aug 26 at 10:02
  • $\begingroup$ The second link has an incorrect answer. It assumes that the length of the time interval in which B can arrive and still meet A is independent of when A arrives. This is not the case. $\endgroup$ – Jaap Scherphuis Aug 26 at 12:56
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The explanation of http://brainstellar.com/puzzles/30 is correct - The original text in their solution was:

Plot the graph of X and Y, where X and Y denote the time each witch arrives. This will form a 60x60 square for total feasible region. The probability that they meet is |X-Y|<= 30. Draw this as favorable region, by joining line (30,0) with (60,30) and (0,30) with (30,60). Clearly the interior of this region has area 3/4 th of total. Hence the probability = 3/4 = 0.75

here it is visually:

2D shaded Graph of Witches arrival times

(the favoured region is the one with 2 hats in it)

The other answer you referenced made basically the following argument:

If Witch 1 arrives after 0 minutes, the other witch can either arrive within 30 minutes (meet) or after 30 minutes (don't meet). These chances are equal (P = 0.5). All other times will have the same probability, so the answer is 0.5.

But if you look at the graph

Same 2D Chart but line W1 = 0 marked

then you can see that

witch 1 arriving at 0 is not typical, but is pretty much the worst case (red line). The best case for witch 1 is arriving after 30 minutes, giving P=1. But it is not enough to look at any single witch 1 arrival, you need to consider them all, which can be done by looking at the area.

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  • $\begingroup$ Damn.. i just wanted to draw the same pic... you were first. +1 $\endgroup$ – TMS Aug 26 at 19:45
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It's:

$75\%$

because:

Imagine witch A arrives at 00:30. Witch B meets witch A with probability $100\%$. Say witch A arrives at 00:00 or 01:00. A meeting occurs $50\%$ of the time (witch B arrives between 00:00-00:30 or 00:30-01:00 respectively). If witch A arrives at say 00:15, then witch B needs to arrive between 00:00-00:45, or $75\%$.

More generally, if witch A arrives at time $t$, witch B must arrive in either the window 00:00-(00:30+t) ($t\le30$) or (t-00:30)-01:00 ($t\ge30$). So the probability is linear in both halves, with the average being $75\%$.

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  • $\begingroup$ Can you please explain "So we can say the probability is linear, with the average being 75%." a bit more? $\endgroup$ – Birjolaxew Aug 26 at 10:05
  • $\begingroup$ @Birjolaxew; If witch A arrives at midnight, there is a 50% chance of a meeting, rising to 100% if witch A arrives at half past, and then decreasing to 50% at 1 o'clock. The chance of a meeting is linear up to half past, and then linear again. (if witch A arrives at 00:10, witches B window is 00:00-00:40 = 66.66%) $\endgroup$ – JMP Aug 26 at 10:11

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