resampling insurance

[afrench77]

I work in laboratory and thinking about selling "resampling insurance" because 1 out of 100 times we make a mistake and the customer must resample. My question is if we make a mistake 1/100 times and 5 out of 100 customers buy the insurance, what are the chances 1 of the 5 people who bought insurance will be the 1 mistake we made in the lab testing?

I will also have to figure out how much to charge for the insurance if I put a cap of $200.00.

Thank you so much for any help!

 

[DrPhil]

I work in laboratory and thinking about selling "resampling insurance" because 1 out of 100 times we make a mistake and the customer must resample. My question is if we make a mistake 1/100 times and 5 out of 100 customers buy the insurance, what are the chances 1 of the 5 people who bought insurance will be the 1 mistake we made in the lab testing?

I will also have to figure out how much to charge for the insurance if I put a cap of $200.00.

Thank you so much for any help!

You have a population probability of \(\displaystyle p=0.01\) that any particular test will be bad. You then make \(\displaystyle n=5\) trials. That gives what is called a "Binomial Distribution." The probability that exactly m will be bad is

\(\displaystyle \displaystyle P(m) = p^m (1-p)^{n-m} \dfrac{n!}{m!\ (n-m)!} \)

A feature of this distribution is that \(\displaystyle p\) does not vary for any sample you take. Thus it is the same for people who buy the insurance as it is for thiose who don't. The average number that will be bad out of 5 trials is just what you would expect, \(\displaystyle \mu = n\ p = 0.05 = 1/20\).

The table of probabilities for 5 trials (calculated with the formula above) is
\(\displaystyle P(0) = 0.951\)
\(\displaystyle P(1) = 0.048\)
\(\displaystyle P(2) = 0.00097\)
\(\displaystyle P(3) = 0.0000098\)
\(\displaystyle P(4) = 0.0000000495\)
\(\displaystyle P(5) = 0.0000000001\)

 

[DrPhil]

I work in laboratory and thinking about selling "resampling insurance" because 1 out of 100 times we make a mistake and the customer must resample. My question is if we make a mistake 1/100 times and 5 out of 100 customers buy the insurance, what are the chances 1 of the 5 people who bought insurance will be the 1 mistake we made in the lab testing?

I will also have to figure out how much to charge for the insurance if I put a cap of $200.00.

Thank you so much for any help!

Aside from my previous post which discusses theoretical stratistics, consider your experience. If it is really true in the long run that you make mistakes 1% of the time, then by charging an additional 1% you would cover the costs of the repeated tests. Your insurance would break even if the premium was 1% (or 2% to cover administrative costs and guarantee you wouldn't lose money).

 

[JeffM]

I work in laboratory and thinking about selling "resampling insurance" because 1 out of 100 times we make a mistake and the customer must resample. My question is if we make a mistake 1/100 times and 5 out of 100 customers buy the insurance, what are the chances 1 of the 5 people who bought insurance will be the 1 mistake we made in the lab testing?

I will also have to figure out how much to charge for the insurance if I put a cap of $200.00.

Thank you so much for any help!

You do not make ONE mistake. The number of mistakes you make is 0.01 * n, where n is the number times you have an opportunity to make a mistake. 1 per 100 does not mean 1.

As Dr. Phil points out, the only opportunities to make mistakes that are relevant to this insurance program are those that apply to people with the insurance. I am reluctant to agree with him that a fee of 1% on your sales price will cover the cost to you of your insurance program. For one thing we do not know what that price is.

So the number of relevant mistakes per sale is 0.01 * 0.05 * n per 0.05 * n sales = 0.01 (on the assumption that insured customers are otherwise similar to non-insured customers).

Your question is ambiguous. Do you mean that at least 1 out of 5 insured sales will have a mistake, at most 1 will have a mistake, or exactly 1 will have a mistake? The answer is different for the three questions.

There are other ambiguities, such as whether you expect most payouts to fall far below $200 or whether you are estimating that payouts will average very close to $200.

If they are very close to $200, you can say, for every five sales to an insured customer

\(\displaystyle P(no\ mistakes) = \dbinom{5}{0} * 0.99^5 * 0.01^0 \approx 0.951.\)

\(\displaystyle P(1\ mistake) = \dbinom{5}{1} * 0.99^4 * 0.01^1 = 0.05 * 0.99^4 \approx 0.048.\)

\(\displaystyle P(2\ mistakes) = \dbinom{5}{2} * 0.99^3 * 0.01^2 = 0.001 * 0.99^3 \approx 0.001\).

\(\displaystyle P(mistakes > 2) \approx 0.\)

So your expected payout will be (approximately) \(\displaystyle 0.951 * 0 + 0.048 * 200 + 0.001 * 400 \approx 10.\)

Now you have to take into account that the probability of more than 2 mistakes in every five sales is in fact slightly greater than zero so the calculation above is slightly to the low side even if all the numbers are exact, which they are not. So you best apply some margin of error. And if you want some profit from this, tack on a little more.

Your major risk is not in the pricing. Your major risk is that the quality of your work goes down and suddenly you are making mistakes 2% or 3% of the time.