Question

3. A homeowner wants to purchase multiple electric pumps and operate them in parallel to ensure that she can handle any basement flooding that occurs. A single electric pump is enough to handle any flooding, but the pumps are not very reliable. She wants to be %99.99 sure that there will be a working pump during a flood.

   1. (a) If she only wants to buy 3 identical electric pumps, what is the required minimum reliability of each pump?

   2. (b) Now suppose the manufacturer specifies each electric pump has a 0.75 reliability. How many would need to be purchased?

   3. (c) After some searching, another more reliable gas-powered pump with the same output capacity as the electric pump was found. These gas-powered pumps have a reliability of 0.85 but cost $1000 dollars each. The electric pumps cost $600. If she is going to buy only one type of pump, what is the cheapest option to meet her reliability goal? What if she is willing to buy both kinds of pumps?

Answer 1

The reliability that needs to be achieved here is 99.99% sure that atleast one of pumps work.

This means we need to buy as many pumps such that the probability that none of the pumps work is less than 0.0001( which is 1-0.9999).

Since each pump working or not working is independent of each other we can multiply the probabilities of not working to reach the probability that none of the pumps work.

a) If she wants to buy only 3 pumps then the probaility of none of them working should be less than 0.0001

Cube root of 0.0001=0.046415888.

We will take 0.0464 as the probability that one pump will not work.

minimum reliability=1-0.0464=0.9536

b)If each has 0.75 reliability that means 0.25 probability of it not working.

We need to see how many times 0.25 multiplied by itself gives less than 0.0001.

0.25*0.25*0.25*0.25*0.25*0.25*0.25( 7 times)=0.00006(less than 0.0001)

Hence 7 pumps needed.

c) Probaility of one gas powered pump not working is 0.15.

So there needs to be 5 gas powered pumps to reach 99.99 reliabilty. Found out by simple multiplication as above.

5*1000=5000

In previous part we found out we need 7 electric ones

7*600=4200

So electric pumps are cheaper for required needs.

There is no combination of the two pumps that would give required reliability of .9999 cheaper than 4200.