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Probability of 3 Events Calculator

1. What is a Probability Calculator (Three Events)?

Definition: This calculator computes the probability of various scenarios involving three independent events A, B, and C.

Purpose: It assists in understanding the likelihood of outcomes in complex experiments or statistical analyses involving multiple events.

2. How Does the Calculator Work?

The calculator uses the following formulas for three independent events:

\( P(A \cap B \cap C) = P(A) \times P(B) \times P(C) \)
\( P(A \cup B \cup C) = P(A) + P(B) + P(C) - P(A) \times P(B) - P(A) \times P(C) - P(B) \times P(C) + P(A) \times P(B) \times P(C) \)
\( P(\text{exactly one}) = P(A) \cdot (1 - P(B)) \cdot (1 - P(C)) + (1 - P(A)) \cdot P(B) \cdot (1 - P(C)) + (1 - P(A)) \cdot (1 - P(B)) \cdot P(C) \)
\( P(\emptyset) = 1 - (P(A) + P(B) + P(C) - P(A) \times P(B) - P(A) \times P(C) - P(B) \times P(C) + P(A) \times P(B) \times P(C)) \)

Steps:

  • Input \( P(A) \), \( P(B) \), and \( P(C) \) as percentages.
  • Select the desired probability scenario from the options.
  • Validate: Probabilities must be between 0% and 100%.
  • Calculate the selected probability using the appropriate formula.
  • Convert to a percentage and round to 4 decimal places.
  • Display the result.

3. Importance of Probability Calculations

These calculations are key for:

  • Complex Experiments: Assessing combined event chances.
  • Statistical Analysis: Predicting multiple event outcomes.
  • Risk Assessment: Evaluating scenarios with multiple risks.

4. Using the Calculator

Examples (P(A) = 50%, P(B) = 30%, P(C) = 20%):

  • All three occurring:
    • \( P(A \cap B \cap C) = 0.5 \times 0.3 \times 0.2 = 0.03 \).
    • Result: 3.0000%.
  • At least one occurring:
    • \( P(A \cup B \cup C) = 0.5 + 0.3 + 0.2 - (0.5 \times 0.3) - (0.5 \times 0.2) - (0.3 \times 0.2) + (0.5 \times 0.3 \times 0.2) = 1 - 0.03 = 0.67 \).
    • Result: 67.0000%.
  • Exactly one occurring:
    • \( P(\text{exactly one}) = (0.5 \times 0.7 \times 0.8) + (0.5 \times 0.3 \times 0.8) + (0.5 \times 0.7 \times 0.2) = 0.28 + 0.12 + 0.07 = 0.47 \).
    • Result: 47.0000%.
  • None of them occurring:
    • \( P(\emptyset) = 1 - 0.67 = 0.33 \).
    • Result: 33.0000%.

5. Frequently Asked Questions (FAQ)

Q: What if probabilities exceed 100%?
A: The calculation is invalid, and an error is displayed.

Q: Are events assumed independent?
A: Yes, the calculator assumes A, B, and C are independent.

Q: What does 'None' mean?
A: It’s the probability that none of the three events occur.

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