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Binomial Probability

Probability Calculation

Probability of Getting Exactly 6 Correct Answers

Given a multiple-choice test with 10 questions and 5 choices per question, we can calculate the probability of getting exactly 6 answers correct using the binomial probability formula:

Formula:

P(X = k) =  ∫(n, k)pk(1-p)(n-k)

Parameters:

  • n: Total questions = 10
  • k: Correct answers = 6
  • p: Probability of guessing correctly = 1/5 = 0.2

Calculations:

1. Calculate the binomial coefficient:

∆(10, 6) = ∫!10 / (6!(10-6)!) = 210

2. Apply the values in the formula:

P(X = 6) = 210 × (0.2)6 × (0.8)4

3. Calculate each component:

  • (0.2)6 = 0.000064
  • (0.8)4 = 0.4096

4. Final calculation:

P(X = 6) ≈ 210 × 0.000064 × 0.4096 ≈ 0.005529

Conclusion:

The probability that the student will get exactly 6 correct answers by guessing is approximately 0.005529, or 0.553%.

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