Question

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

Answer 1

10^-n=10^-6

So n=6

Thus they will want to install 6 switches...

Answer 2

Because every individual switch has a 1/10 chance of failing, two switches will have a (1/10)*(1/10) chance of failing.  Three switches will have a (1/10)*(1/10)*(1/10) chance of failing.  This pattern will continue.  Because $( \frac{1}{10} )^6$ equals 1/1,000,000, 6 switches will give you a 1 in 1 million chance of all failing at the same time.