Certainly! Let's solve the problem step-by-step.
1. Given Information:
- The probability of player A winning the match is [tex]\(\frac{1}{3}\)[/tex].
2. Understanding Probabilities:
- In probability, the sum of the probabilities of all mutually exclusive outcomes must equal 1.
- For this problem, there are only two possible outcomes: either player A wins or player B wins.
3. Setting Up the Equation:
- Let [tex]\(P(A)\)[/tex] be the probability that player A wins the match.
- Let [tex]\(P(B)\)[/tex] be the probability that player B wins the match.
- According to the problem, [tex]\(P(A) = \frac{1}{3}\)[/tex].
4. Calculating the Probability for Player B:
- The sum of the probabilities for player A and player B winning is 1.
- Therefore, we can write the equation:
[tex]\[
P(A) + P(B) = 1
\][/tex]
- We substitute the value of [tex]\(P(A)\)[/tex] from the given information:
[tex]\[
\frac{1}{3} + P(B) = 1
\][/tex]
5. Solving for [tex]\(P(B)\)[/tex]:
- To find [tex]\(P(B)\)[/tex], we solve the equation for [tex]\(P(B)\)[/tex]:
[tex]\[
P(B) = 1 - \frac{1}{3}
\][/tex]
6. Simplifying the Expression:
- Perform the subtraction:
[tex]\[
P(B) = \frac{3}{3} - \frac{1}{3} = \frac{2}{3}
\][/tex]
7. Result:
- Therefore, the probability that player B wins the match is [tex]\(\frac{2}{3}\)[/tex].
So, the probability of winning the match by player B is [tex]\[\frac{2}{3}\][/tex] or approximately 0.6666666666666667.
