In rock-paper-scissors, each player has three choices: rock, paper, or scissors. Assuming each choice is made randomly and independently, the probability of winning a single round against one opponent can be calculated as follows:
- There are three outcomes for each round:
- Win (your choice beats the opponent's choice)
- Lose (your choice is beaten by the opponent's choice)
- Tie (both players make the same choice)
To win a round, you have to choose the winning option against your opponent's choice. There are 3 winning scenarios out of 9 possible outcomes (since there are 3 choices for you and 3 choices for your opponent, leading to \(3 \times 3 = 9\) total combinations). Thus, the probability of winning a single round is:
\[ P(\text{win}) = \frac{3}{9} = \frac{1}{3} \]
If the person plays against five opponents one by one, the probabilities of winning all five rounds would be calculated by multiplying the probability of winning a single round five times:
\[ P(\text{win all 5}) = P(\text{win})^5 = \left(\frac{1}{3}\right)^5 \]
Calculating this:
\[ P(\text{win all 5}) = \frac{1}{3^5} = \frac{1}{243} \]
Thus, the probability of winning all five rounds in a rock-paper-scissors battle against five opponents is:
\[ \boxed{\frac{1}{243}} \]