esra is playing an online multiplayer puzzle game. He has 90% chance of winning any given match, independent of his result on previous matches. if he play two matches, what is the percent probability that he wins them both?

To determine the probability that Esra wins both matches, given that he has a 90% chance of winning any given match independently of the results of previous matches, we follow these steps:

1. **Convert the percentage to a decimal**:
   The probability of Esra winning a single match is 90%, which can be written as 0.90.

2. **Calculate the probability of winning both matches**:
   Since the matches are independent, the probability of winning both matches is the product of the probabilities of winning each match.

   \[
   P(\text{winning both matches}) = P(\text{winning first match}) \times P(\text{winning second match})
   \]

   Given:
   \[
   P(\text{winning first match}) = 0.90
   \]
   \[
   P(\text{winning second match}) = 0.90
   \]

   So:
   \[
   P(\text{winning both matches}) = 0.90 \times 0.90 = 0.81
   \]

3. **Convert the decimal back to a percentage**:
   \[
   0.81 \times 100 = 81\%
   \]

Therefore, the percent probability that Esra wins both matches is 81%.

Would you like more details or have any questions about this solution? Here are five related questions you can ask:

1. What is the probability that Esra loses both matches?
2. How do we calculate the probability of winning at least one match out of two?
3. What if the probability of winning a single match was different, say 70%? How would that change the calculations?
4. How do these calculations change if the matches are not independent?
5. Can we use a similar approach to calculate the probability of winning three matches in a row?

**Tip:** When calculating probabilities for independent events, always multiply the probabilities of each individual event to find the combined probability.

Learn how to calculate the probability of winning two matches in sequence when each match has a 90% chance of winning independently. This problem involves applying the concept of probability for independent events and includes step-by-step calculations. Suitable for high school students learning probability theory.

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