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I am an actuary (FSA 1982) who retired almost 5 years ago who has worked in the field for 43 years. I would have gotten this question easily at the time that I took the test which would have been the Probability and Statistics (Test Number 2) which I passed in Nov 1973. In all of my work I have not needed to perform this type of calculation. Ao not having seen this for 48 years, I did not attempt to solve it. I am sure that I would have been able to do this if I had referenced my Part 2 Book. So I asked my son who is a post-doc PhD, intending on becoming a professor. He was able to come up with the formula in his head and he then put the formula in something that calculates a formula with e (also noted as exp as per DoodleBob) to get the correct answer.

For Mr.Unique and Captain Walker-There are a set of Math Phds such as my son who are committed to math and are involved in math that I would have had no clue about, even in my prime.

So you and DoodleBob did get the correct answer. The mathematical explanation is as follows: “An exponential random variable with mean 2 has a probabilty density function given by (1/2)*e^(-x/2). Therefore, the probabilty that the printer lasts less than 1 year is the integral of this function from 0 to 1 which is 1-e^(-1/2). By self similarity of the eexponential random variable, the probabilty that a printer lasts between 1 and 2 years is the same value multiplied by e^(-1/2). Hence, our answer because we sell 100 printers is
100*{200*(1-e^(-1/2))+100*(e^(-1/2)-e^(-1))}
For those inclined to check e= 2.718281828459... and * means multiplication and ^means exponent (i.e 2^3 is 2 cubed or 8)

Some other thoughts on comments made on this thread (who I did not specifically tag in the names in my reply). The actuarial is a great profession and has provided me with a fine career. In fact, the Actuarial profession is rated among the top professions. There was a time when it was rated as the number 1 profession. When I first started my career, most people were not familiar with the actuarial profession but it has come in prominence particularly with regard to Social Security and Health Care. The Actuarial career has set me up for my greatest position which is retirement. I do not miss the working world one bit as my time is largely my own and I am never bored. I wish you luck and much happiness in your career as an actuary.

One last comment, I appreciate your tag line - particularly from an actuary.

Among other things that was fascinating about this thread (and yes...it IS a fascinating thread), was the WAY how people got to the answer. kosciusko51 simply said D. Neanderthal provided a high-level proof and arrived at D. I stated the answer verbally, proceeded to get the probabilities wrong initially, then made a course correction to arrive at D. Finally, TakeChargeBob gave a technical explanation to arrive at D.

The reason why this is fascinating - or perhaps I'm just a dweeb and it's not fascinating but you're still reading so join the club - is how several people, looking at the same problem, got to the right answer but through similar yet different paths. And yet, this is supposed to be a "science." How can this be?

The answer lies in the Founding of America. Our Founders wished for individuals to be free to pursue their manifest destiny, following clear but limited Constitutional boundaries. Over 200 years later, on a daily basis through an infinite number of choices and decisions, Americans continue to live out that wish. The very fact that several FReepers - allegedly mouth-breathing knuckle draggers per the MSM - took time to solve a risk management problem in various ways, is a testimony to the Founders' desires AND stands as evidence that their wish remains alive.

Oh yes, over that 200+ years Leviathan has encroached increasingly upon that wish. Indeed, we are at the point where SCOTUS will adjudicate in 2022 on the Constitutionality of the executive implementing a law forcing many employers to track their employees shot and testing status, without a legislative vote. Stunning, isn't it?

And yet, the people who think this is a legitimate exercise of force continue to scratch their heads at the opposition. Indeed, in this article a reporter stated breathlessly

And still, more people died from Covid in 2021 than died from Covid in 2020. In 2021, swaths of the country fought mask mandates, opposed vaccination mandates, objected to any measure designed to slow the spread of Covid that they perceived as an impediment on their ability to resume pre-pandemic activities. This insistence on returning to life as normal came at an unfathomable cost — the loss of hundreds of thousands of parents, grandparents, great-grandparents. Aunts and great-aunts, uncles, and great uncles. Cousins. Friends. Coworkers and supervisors. And still, big chunks of the population refused to get vaccinated, refused to wear masks, insisted SARS-2 was a hoax, or was no more threatening than the flu. “It almost is inexplicable,” Anthony Fauci, director of the National Institutes of Allergy and Infectious Diseases, told me when I asked him about this last month.

The reason why these Autnoritarians find this "inexplicable" is because they think there is ONE WAY to "life as normal." And it must be THEIR WAY.

We spent one evening demonstrating that there are many paths to a solution. That Founding spirit IS alive. It cannot be suppressed, even by a legal stroke of the pen.

Thanks for listening. /rant