The probability that a contestant is from France is 0.85, the probability that a contestant scored at least 85 points is 0.78, and the probability that a contestant is from France and scored at least 85 points is 0.69.
What is the probability that a randomly chosen contestant is from France or scored at least 85 points?

To find the probability that a randomly chosen contestant is from France or scored at least 85 points, we can use the formula for the union of two events in probability:

\[
P(A \cup B) = P(A) + P(B) - P(A \cap B)
\]

Here:
- \( P(A) \) is the probability that a contestant is from France, which is 0.85.
- \( P(B) \) is the probability that a contestant scored at least 85 points, which is 0.78.
- \( P(A \cap B) \) is the probability that a contestant is from France and scored at least 85 points, which is 0.69.

Now, substitute the values into the formula:

\[
P(A \cup B) = 0.85 + 0.78 - 0.69
\]

\[
P(A \cup B) = 1.63 - 0.69 = 0.94
\]

So, the probability that a randomly chosen contestant is from France or scored at least 85 points is **0.94**.

Would you like any further details or have any questions? Here are some related questions to consider:

1. What is the probability that a contestant is not from France?
2. What is the probability that a contestant did not score at least 85 points?
3. How would the probability change if \(P(A \cap B)\) were different?
4. Can the probability \(P(A \cup B)\) ever exceed 1? Why or why not?
5. What would \(P(A \cup B)\) be if \(P(A \cap B)\) was 0?

**Tip:** When dealing with probabilities, always ensure that the sum of disjoint probabilities does not exceed 1, as this would violate the basic rules of probability theory.

The probability that a contestant is from France is 0.85, the probability that a contestant scored at least 85 points is 0.78, and the probability that a contestant is from France and scored at least 85 points is 0.69. What is the probability that a randomly chosen contestant is from France or scored at least 85 points?

Calculate the probability that a contestant is either from France or scored at least 85 points using the addition rule of probability. This problem involves key concepts in probability theory and is suitable for students in Grades 10-12.

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