Basic probability confusion A machine has six switches. The probability that any particular switch works properly is $0.98$. Assuming independent operation of the switches, calculate the probability that at least one switch fails to work properly.
Why isn't it this: P(not working) = $1 - 0.98 = 0.02$
There are six switches, so $1/6$ of picking on the switches. We want to find AT LEAST one. Therefore we have $6 * 1/6 * 0.02 = 0.02$
EDIT: I want to know why I am wrong instead of finding the correct answer
 A: The probability that at least one of the switches fails to work properly is $1 - $ the probability that none of the switches fail.  That latter probability is simply $(0.98)^6$, so the sought after probability is $1-(0.98)^6 \approx 0.114958$.
Note that the switches are independent, so the probabilities of each one failing or not multiply.
A: OK, I think I now understand how your thinking went, and therefore where it went wrong:
You are failing to take into account that after testing the first switch, you would not test it again. Also, if the first switch already failed, you don't need to test it again to see that it failed.
Also I don't see the need to test them in random order, you just test them one by one.
So you test the first switch. With a probability of $0.02$, it fails. Then you are ready (you know it failed). Otherwise (that is, with probability $0.98$) you proceed to the second switch. Again, when you test it it will fail with probability $0.02$ and succeed with probability $0.98$. But there's only a probability of $0.98$ that you'll even get to test it, so the probability that you find it to fail is $0.98\cdot 0.2$ and the probability that you find it to succeed is $0.98\cdot 0.98$. In the latter case, you proceed to the third switch, and so on. Unless you've found the last switch to also succeed, one of the switch failed. But the probability that the last switch succeeded is $0.98^6$ because each switch contributed a factor of $0.98$. Since the probabilities add up to $1$, this means that the probability that at least one failed is $1-0.98^6$.
