56. Defective Disks A pack of 100 recordable DVDs contains 5 defective disks. You select four disks. What is the probability of selecting at least three non defective disks?

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Step 1: Identify the total number of disks (N = 100), the number of defective disks (D = 5), and the number of non-defective disks (N - D = 95). You are selecting 4 disks, and the goal is to find the probability of selecting at least 3 non-defective disks.

Step 2: Break the problem into two cases: (1) selecting exactly 3 non-defective disks and 1 defective disk, and (2) selecting all 4 non-defective disks. Use the hypergeometric probability formula for each case: P(X = k) = (C(G, k) * C(B, n - k)) / C(N, n), where G is the number of non-defective disks, B is the number of defective disks, N is the total number of disks, n is the number of disks selected, and k is the number of non-defective disks selected.

Step 3: For case 1 (exactly 3 non-defective disks), calculate the number of ways to choose 3 non-defective disks from 95 and 1 defective disk from 5. This is given by: C(95, 3) * C(5, 1). Then divide by the total number of ways to choose 4 disks from 100: C(100, 4).

Step 4: For case 2 (all 4 non-defective disks), calculate the number of ways to choose 4 non-defective disks from 95. This is given by: C(95, 4). Then divide by the total number of ways to choose 4 disks from 100: C(100, 4).

Step 5: Add the probabilities from case 1 and case 2 to find the total probability of selecting at least 3 non-defective disks. This is P(X = 3) + P(X = 4).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Probability

Probability is a measure of the likelihood that a particular event will occur, expressed as a number between 0 and 1. In this context, it helps determine the chances of selecting a certain number of non-defective disks from a pack. Understanding basic probability principles, such as combinations and the complement rule, is essential for solving problems involving random selections.

Combinations

Combinations refer to the selection of items from a larger set where the order does not matter. In this scenario, we need to calculate how many ways we can choose non-defective disks from the total available. The formula for combinations, denoted as C(n, k) = n! / (k!(n-k)!), is crucial for determining the number of successful outcomes in probability calculations.

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 question, calculating the probability of selecting at least three non-defective disks can be simplified by first finding the probability of selecting fewer than three non-defective disks and subtracting that from one. This approach often makes complex probability problems more manageable.