A sample of 4 different calculators is randomly selected from a group containing 11 that defective and 25 that have no defects. What is the probability that at least one of the calculators is defective?

To find the probability that at least one of the calculators is defective, we can use the complement rule. The complement of "at least one calculator is defective" is "none of the calculators are defective." Let's calculate the probability of none of the calculators being defective first, and then use the complement rule to find the probability of at least one calculator being defective. 1. Calculate the probability of none of the calculators being defective: - Total number of calculators = 11 (defective) + 25 (no defects) = 36 - Probability of selecting a non-defective calculator = 25/36 - Probability of selecting 4 non-defective calculators = (25/36) * (24/35) * (23/34) * (22/33) - Probability of none of the calculators being defective = (25/36) * (24/35) * (23/34) * (22/33) 2. Calculate the probability of at least one calculator being defective: - Probability of at least one calculator being defective = 1 - Probability of none of the calculators being defective Let's calculate these probabilities. Calculate the value by following steps: - step0: Calculate: \(\frac{25}{36}\times \frac{24}{35}\times \frac{23}{34}\left(\frac{22}{33}\right)\) - step1: Reduce the fraction: \(\frac{25}{36}\times \frac{24}{35}\times \frac{23}{34}\times \frac{2}{3}\) - step2: Multiply the terms: \(\frac{10}{21}\times \frac{23}{34}\times \frac{2}{3}\) - step3: Multiply the terms: \(\frac{115}{357}\times \frac{2}{3}\) - step4: Multiply the fractions: \(\frac{115\times 2}{357\times 3}\) - step5: Multiply: \(\frac{230}{1071}\) The probability of none of the calculators being defective is approximately 0.214753. Now, let's calculate the probability of at least one calculator being defective using the complement rule. Calculate the value by following steps: - step0: Calculate: \(1-0.214753\) - step1: Subtract the numbers: \(0.785247\) The probability that at least one of the calculators is defective is approximately 0.785247 or 785247/1000000.