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59. Two players, [tex]A[/tex] and [tex]B[/tex], play a tennis match. If the probability of winning the match by [tex]A[/tex] is [tex]\frac{1}{3}[/tex], what is the probability of winning the match by [tex]B[/tex]?

(Note: The original text included non-English characters that were not part of the question and were thus removed for clarity.)



Answer :

GinnyAnswer
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.

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