There are 25 calculators in a box. Two of them are defective. Suppose 5 calculators are randomly picked for inspection (i.e., each has the same chance of being selected), what is the probability that only one of the defective calculators will be included in the inspection?a)1/2b)1/3c)1/4d)1/5Correct answer is option 'B'. Can you explain this answer?

Question

Answer

Probability of only one is defective out of 5 calculators

Answer

To solve this problem, we need to use the concept of probability.

Step 1: Calculate the total number of ways to select 5 calculators out of 25
The total number of ways to select 5 calculators out of 25 is given by the combination formula:

C(25, 5) = 25! / (5! * (25-5)!) = 53130

So, there are 53130 different ways to select 5 calculators from the box.

Step 2: Calculate the number of ways to select only one defective calculator
To calculate the number of ways to select only one defective calculator, we need to consider two cases:

Case 1: Selecting one defective calculator and four non-defective calculators
The number of ways to select one defective calculator out of two is given by the combination formula:

C(2, 1) = 2! / (1! * (2-1)!) = 2

The number of ways to select four non-defective calculators out of the remaining 23 is given by the combination formula:

C(23, 4) = 23! / (4! * (23-4)!) = 8855

So, the total number of ways to select only one defective calculator is 2 * 8855 = 17710.

Case 2: Selecting one non-defective calculator and one defective calculator and three non-defective calculators
The number of ways to select one non-defective calculator out of 23 is given by the combination formula:

C(23, 1) = 23! / (1! * (23-1)!) = 23

The number of ways to select one defective calculator out of two is given by the combination formula:

C(2, 1) = 2! / (1! * (2-1)!) = 2

The number of ways to select three non-defective calculators out of the remaining 22 is given by the combination formula:

C(22, 3) = 22! / (3! * (22-3)!) = 1540

So, the total number of ways to select only one defective calculator is 23 * 2 * 1540 = 70760.

Step 3: Calculate the probability
The probability of selecting only one defective calculator is given by:

P(only one defective calculator) = Number of ways to select only one defective calculator / Total number of ways to select 5 calculators

P(only one defective calculator) = (17710 + 70760) / 53130

P(only one defective calculator) = 88470 / 53130

P(only one defective calculator) ≈ 1/3

Therefore, the probability that only one of the defective calculators will be included in the inspection is approximately 1/3.

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