Themed Challenge 2 (Manga/Anime): The Exhaustive Battle

Probability Level 4

Pikachu and a wild Rattata have been battling for a long time. They have been fighting for so long that their accuracy is haywire, and their health is one on-target hit away from being a K.O. Any attack now, however powerful or weak, if on target, could defeat the opponent.

At this stage of battle, Rattata has 2 2 attacks. He can use Tackle, which has a 1 3 \frac{1}{3} chance of being on target. He can also use Hyper Fang, but only after 2 2 moves have occurred (i.e. Rattata moves, then Pikachu moves, then Rattata can use it or save for a later move). This attack has a 2 3 \frac{2}{3} accuracy rate.

Pikachu, on the other hand has 3 3 attacks. One is Quick Attack, which has a 1 3 \frac{1}{3} probability of being on target. Another attack is ThunderShock, which has a 1 2 \frac{1}{2} probability of being on target, but can only be used after 2 2 moves. Finally, Pikachu can use ShockWave, a move with Infinite Accuracy, which, mathematically speaking has an accuracy rate of 1 1 . However, it can only be used after 4 4 moves.

In this battle, the highest probability of Rattata winning and defeating Pikachu is given by M N \frac{M}{N} , where M , N M,N are co-prime integers. Find M + N M+N .


The answer is 109.

Nanayaranaraknas Vahdam by Nanayaranaraknas Vahdam , 22, India

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1 solution

The probability of Rattata winning is comprehensively the possibility of its survival in the first two rounds, and the probability of it beating Pikachu in the third round. It is necessary that Rattata beat Pikachu before the third round, otherwise Pikachu can use ShockWave, and immediately defeat Rattata. The possibility of survival in the first round can be divided into two parameters: whether his attack is on target, or whether both Rattata and Pikachu miss. Thus, the survival in the first round for Rattata is given by 1 3 + 4 9 \frac{1}{3} + \frac{4}{9} = = 7 9 \frac{7}{9} . In the second round, for highest probability of a hit Pikachu will use ThunderShock, because regardless of what move Pikachu makes in the second round, it will use ShockWave in the third round, and finish the battle. Now, Rattata can either use Tackle or Hyper Fang.

Case: 1 Rattata uses Hyper Fang

The probability of survival in the second round is given by 2 3 + 1 6 \frac{2}{3} + \frac{1}{6} = = 5 6 \frac{5}{6} . Now, in the third round, he can only use Tackle. Thus, probability of survival in the third round is just 1 3 \frac{1}{3} .

Thus, total possibility of survival for Rattata is 7 9 × 5 6 × 1 3 \frac{7}{9}\times\frac{5}{6}\times\frac{1}{3} = = 35 162 \boxed{\frac{35}{162}}

Case: 2 Rattata uses Tackle

The probability of survival in the second round is given by 1 3 + 1 3 \frac{1}{3} + \frac{1}{3} = = 2 3 \frac{2}{3} . Now, in the third round, he can use Hyper Fang. Thus, probability of survival in the third round is just 2 3 \frac{2}{3} .

Thus, total possibility of survival for Rattata is 7 9 × 2 3 × 2 3 \frac{7}{9}\times\frac{2}{3}\times\frac{2}{3} = = 28 81 \boxed{\frac{28}{81}}

Thus, the highest possibility of survival is 28 81 \frac{28}{81} . Thus, the answer is 109 \boxed{109}

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