Am I using the concept of expected value correctly in the following thought experiment?
I have a machine. I currently believe that there is a 50-50 chance it is broken. I know that buying a new one will cost $329.
If I buy a new one right now, there’s no chance I’ll get evidence out of thin air that the old one is actually fine so the expected value is $329*100%=$329.
If I spend $5.45, I can get enough evidnece to either put an end to the question or give me a 70% chance that the machine is really broken. Now the expected value calculation is $5.45*.5 = $2.73 if the machine was fine all along or ($329+5.45)*.5=$167.23 if I race off and buy a new machine. Adding the two, I get $169.96.
Finally, I can spend $75 on a fancy test that will conclusively say if the machine is broken. The expected value for “machine was fine all along” if I get this far is ($5.45+$75) * .3=$24.14 or $286.62 if my machine turned out to need replacing with a total cost of $310.76.
It feels right to me that the cost of doing the simple test is a much smaller number than assuming the machine is broken but is it really right to say that the cost of all gathering all the extra evidence & buying a new machine is still cheaper than buying the machine without doing the extra testing?
Extra conjecture: The answer is “Yes” because if I do all the extra work, there’s lots of chances for me to branch off into the outcome of “Oh, that machine was fine all along” and stop right there.
Am I using the concept of expected value correctly in the following thought experiment?
I have a machine. I currently believe that there is a 50-50 chance it is broken. I know that buying a new one will cost $329.
If I buy a new one right now, there’s no chance I’ll get evidence out of thin air that the old one is actually fine so the expected value is $329*100%=$329.
If I spend $5.45, I can get enough evidnece to either put an end to the question or give me a 70% chance that the machine is really broken. Now the expected value calculation is $5.45*.5 = $2.73 if the machine was fine all along or ($329+5.45)*.5=$167.23 if I race off and buy a new machine. Adding the two, I get $169.96.
Finally, I can spend $75 on a fancy test that will conclusively say if the machine is broken. The expected value for “machine was fine all along” if I get this far is ($5.45+$75) * .3=$24.14 or $286.62 if my machine turned out to need replacing with a total cost of $310.76.
It feels right to me that the cost of doing the simple test is a much smaller number than assuming the machine is broken but is it really right to say that the cost of all gathering all the extra evidence & buying a new machine is still cheaper than buying the machine without doing the extra testing?
Extra conjecture: The answer is “Yes” because if I do all the extra work, there’s lots of chances for me to branch off into the outcome of “Oh, that machine was fine all along” and stop right there.