A and B play a game with two others individually. The probability of A wining the game is 1/3 and
that of Bis 2/5 Find the probability that only one of them is winner.

The problem asks for the probability that either A or B wins the game but not both. Let’s break it down step-by-step.

### Notation:
- Probability of A winning the game, \( P(A) = \frac{1}{3} \)
- Probability of B winning the game, \( P(B) = \frac{2}{5} \)
- The probability of A **not** winning, \( P(A^c) = 1 - P(A) = 1 - \frac{1}{3} = \frac{2}{3} \)
- The probability of B **not** winning, \( P(B^c) = 1 - P(B) = 1 - \frac{2}{5} = \frac{3}{5} \)

We want to find the probability that only one of them wins, i.e., either A wins and B loses, or B wins and A loses.

### Case 1: A wins, B loses
The probability for this scenario is \( P(A) \times P(B^c) \):
\[
P(A \text{ wins and } B \text{ loses}) = P(A) \times P(B^c) = \frac{1}{3} \times \frac{3}{5} = \frac{1}{5}
\]

### Case 2: B wins, A loses
The probability for this scenario is \( P(B) \times P(A^c) \):
\[
P(B \text{ wins and } A \text{ loses}) = P(B) \times P(A^c) = \frac{2}{5} \times \frac{2}{3} = \frac{4}{15}
\]

### Total Probability
Since these two cases are mutually exclusive (either A wins and B loses or B wins and A loses), we can add the probabilities:
\[
P(\text{Only one wins}) = \frac{1}{5} + \frac{4}{15}
\]
To add these, we need a common denominator:
\[
\frac{1}{5} = \frac{3}{15}
\]
So,
\[
P(\text{Only one wins}) = \frac{3}{15} + \frac{4}{15} = \frac{7}{15}
\]

Thus, the probability that only one of them wins is \( \frac{7}{15} \).

Would you like further clarification or more details?

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Here are 5 related questions:
1. What is the probability that neither A nor B wins?
2. What is the probability that both A and B win?
3. If the probability of A winning increased to \( \frac{1}{2} \), how would the result change?
4. How do probabilities change if A and B are dependent events?
5. What is the complementary probability that both either win or lose together?

**Tip:** Probabilities are always between 0 and 1. When you add probabilities, make sure the events are mutually exclusive!

Calculate the probability that only one of two players, A and B, wins a game with given probabilities of winning. Understand the concepts of probability and mutual exclusivity.

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