If a safety switch has a 1 in a 10 chance of failing how many switches would a company want to install in order to have only a 1 in one million chance of them all failing ay the same time

It is given that one safety switch has a probability of 1/10 of failing. The question asks how many switches the company should install so that the probability of all the switches failing at the same time is 1/1,000,000.

Failing AT THE SAME TIME, means that the chosen switch will have probabilities 1/10 and 1/10 and 1/10 and 1/10
AND means multiply, therefore
1/10 * 1/10 *1/10 * 1/10 * 1/10 *1/10 = 1/1,000,000

This is also modeled by the equation:
(1/10)^n = 1/1,000,000 
n = 6 safety switches.