A warehouse in Andijan stores electronics, and its insurance company set a new condition: the fire-alarm system must work, when needed, with a probability of at least 98 percent.
The manager, Sardor, received two quotes. The first was a premium alarm that reliably works 95 percent of the time and costs 1,200 dollars. The second was a pair of cheaper alarms, each of which works 90 percent of the time, at 450 dollars each, or 900 dollars for both.
At first the premium alarm looked safer: 95 is more than 90. But Sardor remembered how to think about at least one of them. The system with two alarms fails only if both alarms fail at the same moment. Each one fails 10 percent of the time, so both fail together one tenth of one tenth of the time, which is 1 time in 100. The pair therefore works 99 percent of the time.
The premium alarm fails 5 times in 100. The cheaper pair fails 1 time in 100, and it costs 300 dollars less. Only the pair met the insurance company's 98 percent rule.
There was one condition that Sardor almost missed. The calculation is correct only if the two alarms fail independently. If both run on the same power line, a single power cut silences both at once, and the 1 in 100 figure becomes meaningless. He asked the installer to give each alarm its own battery and to mount them in different parts of the building.
The insurance inspector approved the system the following week. Her report used the same arithmetic, and she added one line Sardor liked: two independent ninety-percent systems are better than one ninety-five-percent system.