In Exercises 49-53, use counting principles to find the probability.
51. A shipment of 200 calculators contains 3 defective units. What is the probability that a sample of three calculators will have
c. at least one defective calculator?

Verified step by step guidance

1. Step 1: Understand the problem. We are tasked with finding the probability that a sample of three calculators, chosen from a shipment of 200 calculators (3 of which are defective), contains at least one defective calculator. This is a complementary probability problem, so we will first calculate the probability of the complement event (no defective calculators in the sample) and subtract it from 1.
2. Step 2: Define the complement event. The complement event is that all three calculators in the sample are non-defective. There are 200 calculators in total, and 200 - 3 = 197 of them are non-defective. The probability of selecting only non-defective calculators can be calculated using combinations.
3. Step 3: Use combinations to calculate the total number of ways to choose 3 calculators from the shipment. The total number of ways to choose 3 calculators from 200 is given by the combination formula: C(n, r) = n! / [(r!)(n - r)!]. Here, n = 200 and r = 3. Compute C(200, 3).
4. Step 4: Use combinations to calculate the number of ways to choose 3 non-defective calculators from the 197 non-defective ones. This is given by C(197, 3), where n = 197 and r = 3. Compute C(197, 3).
5. Step 5: Calculate the probability of the complement event (no defective calculators) by dividing the number of ways to choose 3 non-defective calculators by the total number of ways to choose 3 calculators: P(no defective) = C(197, 3) / C(200, 3). Finally, subtract this probability from 1 to find the probability of at least one defective calculator: P(at least one defective) = 1 - P(no defective).

Key Concepts

Probability

Probability is a measure of the likelihood that an event will occur, expressed as a number between 0 and 1. In this context, it quantifies the chance of selecting at least one defective calculator from a sample of three. Understanding how to calculate probability is essential for solving problems involving random selections from a finite population.

Counting Principles

Counting principles, such as combinations and permutations, are mathematical techniques used to determine the number of ways to select items from a larger set. In this problem, combinations are particularly relevant as we need to calculate the different ways to choose calculators from the shipment, which helps in determining the total outcomes for the probability calculation.

Complement Rule

The complement rule in probability states that the probability of an event occurring is equal to 1 minus the probability of it not occurring. In this case, to find the probability of selecting at least one defective calculator, it is often easier to first calculate the probability of selecting none and then subtracting that from 1. This approach simplifies the calculation process.