I figured this puzzle out by going through all the steps the long way. The idea is:
- Find all the times where the hands are equally far from the 12, and calculate the intervals between each one
- Sum up the intervals according to who would win the bet at this time interval
Let's call the angle of the hour hand $a$ and the angle of the minute hand $b$. The angles will be expressed in fractions of one revolution.
For a given time (hour, minute or $h$:$m$) we can calcluate the hand angles:
\begin{align} b &= \frac{m}{60} \\ a &= \frac{h}{12} + \frac{\frac{m}{60}}{12} \\ &= \frac{h}{12} + \frac{b}{12} \end{align}
eg. for the time $11$:$15$, $h=11$ and $m=15$ so $b=\frac{1}{4}$ and $a=$ some fraction getting close to $1$.
There are two ways for the hands to be equally far from $12$. Either on the same side, or opposite sides (like the OP's picture). These times are represented by $a = b$ and $1 - a = b$.
First, $a = b$, by substitution
\begin{align} h / 12 + b / 12 &= b \\ h &= 11b \\ \text{and} \\ m &= 60b \end{align}
So, for any given $h$, we can find the $b$ then find $m$ for it. Knowing that the hands cross at most once per hour we can find for $h=0..12$, these are the times when the hands cross.
h b
(0, 0) 12:00
(1, 1/11) 1:(60/11)
(2, 2/11) 2:(120/11)
(3, 3/11) 3:(180/11)
(4, 4/11) 4:(240/11)
(5, 5/11) 5:(300/11)
(6, 6/11) 6:(360/11)
(7, 7/11) 7:(420/11)
(8, 8/11) 8:(480/11)
(9, 9/11) 9:(540/11)
(10, 10/11) 10:(600/11)
(11, 1) 12:00
Now, do the same for when $1 - a = b$,
\begin{align} 1 - \frac{h}{12} - \frac{b}{12} &= b \\ 12 - h &= 13b \\ -h &= 13b - 12 \\ h &= -13b + 12 \\ b &= \frac{12 - h}{13} \end{align}
So again we have a formula relating $b$ to $h$, and we use $h$ to find $b$. Then use $b$ to find $m$. This gives the time in $h$:$m$ where the hands are in an 'iscoscoles' shape.
h b
(0, 12/13) 12:(60*12/13)
(1, 11/13) 1:(60*11/13)
(2, 10/13) 2:(60*10/13)
(3, 9/13) 3:(60*9/13)
(4, 9/13) 4:(60*8/13)
(5, 7/13) 5:(60*7/13)
(6, 6/13) 6:(60*6/13)
(7, 5/13) 7:(60*5/13)
(8, 4/13) 8:(60*4/13)
(9, 3/13) 9:(60*3/13)
(10, 2/13) 10:(60*2/13)
(11, 1/13) 11:(60*1/13)
(12, 0) 12:00
Now let's change both of those sets of times into proper fractions with a common denominator,
Crossing 'x':
(0, 0) 12:00
(1, 1/11) 1:(780/143)
(2, 2/11) 2:(1560/143)
(3, 3/11) 3:(2340/143)
(4, 4/11) 4:(3120/143)
(5, 5/11) 5:(3900/143)
(6, 6/11) 6:(4680/143)
(7, 7/11) 7:(5460/143)
(8, 8/11) 8:(6240/143)
(9, 9/11) 9:(7020/143)
(10, 10/11) 10:(7800/143)
(11, 1) 12:00
Isosceles shape 'v':
(0, 12/13) 12:(7920/143)
(1, 11/13) 1:(7260/143)
(2, 10/13) 2:(6600/143)
(3, 9/13) 3:(5940/143)
(4, 9/13) 4:(5280/143)
(5, 7/13) 5:(4620/143)
(6, 6/13) 6:(3960/143)
(7, 5/13) 7:(3300/143)
(8, 4/13) 8:(2640/143)
(9, 3/13) 9:(1980/143)
(10, 2/13) 10:(1320/143)
(11, 1/13) 11:(660/143)
(12, 0) 12:00
Now let's actually put the times into the order they occur and label them with x
or v
. Also label whether the interval is 'winning' for the short hand or the long hand, which alternates although note the two v
's which switch it up in the middle.
(0, 0) 12:00 x +7920 short
(0, 12/13) 12:(7920/143) v +660 +780 long
(1, 1/11) 1:(780/143) x +6480 short
(1, 11/13) 1:(7260/143) v +1320 +1560 long
(2, 2/11) 2:(1560/143) x +5040 short
(2, 10/13) 2:(6600/143) v +1980 +2340 long
(3, 3/11) 3:(2340/143) x +3600 short
(3, 9/13) 3:(5940/143) v +2640 +3120 long
(4, 4/11) 4:(3120/143) x +2160 short
(4, 9/13) 4:(5280/143) v +3300 +3900 long
(5, 5/11) 5:(3900/143) x +720 short
(5, 7/13) 5:(4620/143) v +3960 +3960 long
(6, 6/13) 6:(3960/143) v +720 short
(6, 6/11) 6:(4680/143) x +3900 +3300
(7, 5/13) 7:(3300/143) v +2160
(7, 7/11) 7:(5460/143) x +3120 +2640
(8, 4/13) 8:(2640/143) v +3600
(8, 8/11) 8:(6240/143) x +2340 +1980
(9, 3/13) 9:(1980/143) v +5040
(9, 9/11) 9:(7020/143) x +1560 +1320
(10, 2/13) 10:(1320/143) v +6480
(10, 10/11) 10:(7800/143) x +780 +660
(11, 1/13) 11:(660/143) v +7920
(11, 1) 12:00 x
Now it's a task of actually summing the numbers and that will give you a value $n$, where $\frac{n}{143}$ is the amount of minutes in a $12$ hour period. To calculate this I went through and calculated the minutes, each time an hour ticked over, I added the amount of minutes left in that hour. It's tedious and I didn't realise a nice way to do this yet.
The total amount of time that the short hand leads is $\frac{51840}{143}$ minutes every $12$ hours and the long hand is $\frac{51120}{143}$ minutes. $51840:51120$ reduces to $72:71$ giving the chances of each outcome at $\frac{72}{143}$ and $\frac{71}{143}$.