4

Two players take turns shooting at a target, with each shot by player $i$ hitting the target with probability $p_i$, $i=1,2$. Shooting ends when two consecutive shots hit the target. What is the probability that the player who shoots first will win?

I understand this problem on a more simple level, when the win condition is only one hit, although I do not know how to solve it given two consecutive hits.

Brian61354270
  • 2,170
Thief
  • 41
  • not clear which rules applies: title and body seem to specify different rules. Please clear – G Cab Mar 06 '20 at 22:12
  • 1
    Specifically: If a player hits, does he take another shot immediately or does he wait while the other player shoots? – herb steinberg Mar 06 '20 at 22:15
  • They take their shots in turns, so person 1 shoots first with a probability of p1 and then person 2 shoots with a probability of p2 and then it repeats until one of them has hit two consecutive shots – Thief Mar 06 '20 at 22:16
  • This is a deceptively difficult problem. I'm interested to see if anyone can come up with an elegant solution. – Math1000 Mar 07 '20 at 00:39
  • It is not an elegant approach: Considering the most recent $2$ shots (i.e. each player last shot) as the states, we can construct a markov chain with $8$ transient state (hit-miss combination consists of $2 \times 2 = 4$, and player $1$, $2$ turns double it) and $2$ absorbing state (player $1$ or $2$ wins). So the required probability is just the absorbing probability in player $1$ winning state. – BGM Mar 07 '20 at 15:04

0 Answers0