Luke The Cat 2
Based on the original Luke the cat problem by Yuxuan Seah
Luke the cat spies a mouse 10m away. Luke travels at 4 metres per second while the mouse is running away at 2 metres per second. When Luke reaches the mouse, he will start swiping at it once every second, while maintaining a speed equal to the mouse. However, Luke is tired and hungry and can only kill 1/3 of the time. He will just cripple the mouse 1/3 of the time and miss completely 1/3 of the time. If he cripples the mouse, his next swipe will kill without fail. The mouse hole is 24m away from Luke. The probability that Luke kills the mouse is a/b, where a and b are coprime integers. Find ab.
Note: Luke can attack the instant he catches up with the mouse, and can attack again on the second that the mouse dives into his hole.
The answer is 675.
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1 solution
In five seconds, Luke will reach the cat. There is no variance at this stage. Once Luke reaches the cat, he has 3 seconds to make his kill, since he will have travelled 20 metres within that time and then the distance from the hole will be 4m. It is 3 seconds because the mouse is travelling at 2m/second and will reach his hole in 2 seconds. Then, the cat can still swipe at the mouse one second after he reaches his hole. If he cripples the mouse or kills it outright on the first second, then it will be a kill. Therefore, he must miss on the first shot for any chance for the mouse to get away. the chance of him missing is 1/3. Again, if he kills or cripples the mouse, it will definitely be his kill. He must again miss for a chance for the mouse's freedom. The chance of this is 1/9 (1/3 x 1/3= 1/9.) This will bring us to the last second. He only has 1/27 chance to kill at this stage (1/3 for kill x 1/9,) for if he cripples the mouse, it will still get away as he does not have the extra second for him to kill it outright and missing has no effect. Therefore, the mouse has a 2/27 chance of getting away and the cat must then have a 25/27 chance of a kill. To find ab, we must do 25 x 27 to give us 675.