False positives and false negatives

When testing for the presence of a disease, the results obtained are often not 100% reliable. In this article we explain false positives and false negatives in test results. The graphics are dynamic, if you change the input numbers, then they'll update.
Let's start with the prevalence of disease, this is the proportion of positive cases in the population.
Disease prevalence input: %


Light red shows the proportion of the population that have the disease, i.e. are positive.
Light blue shows the proportion of the population that don't have the disease, i.e. are negative.


The false negative rate is the proportion of the positive cases that are incorrectly deemed to be negative.
False negative input: %


Dull red shows the true positives: the proportion of the population that are positive and get positive test results.
Orange shows the false negatives: the population that are positive but get negative test results.


The false positive rate is the proportion of the negative cases that are incorrectly deemed to be positive.
False positive input: %


Yellow shows the false positives: the population that are negative but get positive test results.
Dull blue shows the true negatives: the proportion of the population that are negative and get negative test results.


One challenge when considering an individual positive test result is that it is not clear whether it is a true positive or a false positive. We can look at the likelihood of each:

Dull red shows the true positives: the proportion of the population that are positive and get positive test results.
Yellow shows the false positives: the population that are negative but get positive test results.

Using the values that were input above for prevalence, false positives and false negative,
we find that of the positives:

True positives:
False positives:

When the prevalence is low, it is common to have a situation where there are more false positives than true positives. You can try this out for yourself by reducing the prevalence number above.

The Mathematics of a Ponzi scheme

A Ponzi Scheme is a form of investment where profits are paid to early investors with funds from later investors. It often involves fraud and as a result it is of course illegal.
This article will explore the mathematics of a Ponzi Scheme which has been transparent about its intentions. Suppose someone were to set up a fund that aims to pay investors a return of \(\rho\) (rho).
I.e. for each unit of currency invested, it will aim to return \(1+\rho\).
The fund pays out on a first in, first out basis. Early investor money is returned when (or if) later money arrives.
Suppose, at a point in time, the cumulative money invested is N.
The money that's been paid out is P and the fund has remaining money R.
We known that
(Investment in) = (Payouts made) + (Remaining money)
\[N = P + R\] We can classify the investment so far into two categories. There is the money that has already been settled S, i.e. the early investors that have already received their funds returned with a positive return \(\rho\) .
And there is the balance outstanding B, that remains unsettled as yet.
So we can write:
(Investment in) = (Settled amount) + (Balance outstanding) \[N = S + B\]
The early investors who have received their settlement all got a return of \(\rho\).
So we know that the payout: \[P=S(1+\rho)\] The fund administrator has guaranteed that the maximum loss that could be imposed on an investor is \(\lambda\) (lambda).
So for each unit of currency invested, the worst case scenario for the investor is that \((1-\lambda )\) is returned.
Hence we require the remaining funds (R) to be able to cover a payment of the balance (B) with an imposed loss. \[R= B( 1 - \lambda )\] We now have 4 equations and 4 unknowns (P,R,S,B) along with three knowns (N, \(\rho\), \(\lambda\) ).
If we do the algebra, we can find our unknows:

Settled amount:	\(S = N \frac{\lambda}{\lambda + \rho} \)
Balance outstanding:	\(B=N-S\)
Payouts made:	\(P=S (1+ \rho) \)
Remaining money:	\(R=N-P \)

We can try that out with some numbers:

Inputs
Total invested funds (N)		units of currency
Investment return (\(\rho\))		%
Acceptable loss (\(\lambda\))		%
Outputs
Settled amount (S)	-
Balance outstanding (B)	-
Payouts made (P)	-
Remaining money (R)	-

With such a scheme, any investor would be allowed to demand an immediate repayment of funds at any time, though in that case a loss of \(\lambda\) would be imposed. In other words, for each unit of currency invested, \(1-\lambda\) would be returned.

One problem for early investors is that they don't know how long they will have to wait until they'll be repaid their investment with the positive return (\(\rho\)). But it is not just a question of when they'll be paid, it is uncertain if they'll be paid. When later investors don't arrive, the early investors just wait and wait, until they eventually give up and request a return of funds, in that case they'll just have to accept a loss.

It is also worth noting that this model above ignores any administrative fees that may be imposed on funds on the way in and or the way out.