I am reading A First Course in Probability by Sheldon Ross and get stuck by this expression:

The company sells the screws in packages of 10 and offers a money-back guarantee that at most 1 of the 10 screws is defective.

In my understanding, it means if 1 or more than 1 screws are defective the package should be delivered back to the company. But I am wrong because it excludes the 1.


Notice that the number of defective screws in a package is one of {0, 1, 2, 3,..., 10}. The guarantee is that "at most 1 of the 10 screws is defective." This means the most that the guarantee allows is 1. In other words, the company guarantees that 0 screws or 1 screw is defective in the package, and you will not be offered your money back in this case. Otherwise, there are 2 or more defective screws in the package and the company will offer your money back.

In mathematical notation, let G be the number of defective screws in the package. If G≤1, then you do not get your money back. Otherwise, G>1 ⇒ G≥2, and the company offers your money back.

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This means that if more than 1 screw in 10 (probably if 2 or more in any package of 10) are defective, the company will pay on the guarantee. If only 1 in 10 is defective, the company will not pay.

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