A machine consists of three components, which are components X, Y, and Z. These three components operate properly independently of one another. The machine operates properly only when component X operates properly and at least one of components Y and Z operates properly. The probability that component X operates properly is 0.8, the probability that component Y operates properly is 0.4, and the probability that component Z operates properly is 0.3. What is the probability that the machine operates properly?

A) 0.096
B) 0.252
C) 0.464
D) 0.583
E) 0.648

In all the Kaplan explanations for probability questions, it always says, "It is easier for us to find the probability of XYZ not occurring".

Can anyone explain why that is? Why can't we just find the probability of XYZ actually occurring?

For example, in the question above, they tell us outwardly the probability of Components Y and Z operating properly, 0.4 and 0.3 respectively. My inclination is to simply multiply the probability of Component X working (0.8) with the probabilities of Y and Z working, 0.8 * 0.4 * 0.3 = 0.096. While, I know that sounds unreasonably low and is ultimately the incorrect answer, why doesn't this simple approach work?

The explanation states that you must first find the probability of Y and Z NOT operating properly, i.e.

1 - 0.4 = 0.6
1 - 0.3 = 0.7

Then multiply them to get 0.42 and subtract it from 1 to find the probability of either Y or Z operating properly, which is 0.58. At that point you just multiply 0.58 by 0.8 and the answer is 0.464.

If anyone can shed light on this, I'd appreciate it. Thanks everyone!