Samhita faces a three-headed dragon . She wins a 'Tactical medal' if she manages to defeat exactly one of the three heads. The battle proceeds head-by-head under the following conditions: The probability of defeating the first head is 1 3 . After a win: if she defeats a head, the probability of defeating the next head is 2 3 . After a loss: if she fails to defeat a head, the probability of defeating the next head is

Step-by-step solution

Let W denote defeating a head (win) and L denote failing to defeat a head (loss). The probability of exactly one win in three battles is the sum of the probabilities of the sequences WLL , LWL , and LLW . From the given conditions, the initial probabilities are: P(W₁) = 1 3 P(L₁) = 1 - 1 3 = 2 3 The transition probabilities after a win are: P(W_ i+1 |W_i) = 2 3 P(L_ i+1 |W_i) = 1 - 2 3 = 1 3 The transition probabilities after a loss are: P(W_ i+1 |L_i) = 1 4 P(L_ i+1 |L_i) = 1 - 1 4 = 3 4 Calculating the probabilit