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I would like to calculate the value of bacteria on 4 surfaces $i=\{1..4\}$. A person touches some of those 4 surfaces at random and a count is made on their finger after each surface contact ($x_i$).

Someone lost the bacteria count ($x_i$) after each surface but I do know the total count (X) on a person's finger after they've touched a number of surfaces. I also know which ones and in which order.

What I know:

  1. Final bacteria count on a person's finger: $X$
  2. Transfer efficiency from surface to finger: $PT_i=\displaystyle \frac{\text{Finger contact area}}{\text{Area of surface}_i}\frac{1}{\gamma_i}$ where $\gamma$ is a surface dependent constant.
  3. The number of times the person touched a particular surface: $h_i$.

If I had surface counts $C_i$, the summation of bacteria is linear: ie $\begin{eqnarray} h_1C_1PT_1&=&x_1\\ h_2C_2PT_2&=&x_2\\ \vdots\quad &=& \quad \vdots\\ h_iC_iPT_i&=&x_i \end{eqnarray}$

such that summing over all surfaces $i$ the total count x is: $\displaystyle \sum_i h_iC_iPT_i=\sum_i x_i=X$.

Can I back calculate $C_i$, without $x_i$ even statistically?
Best regards.

ANSWER?

so since the portion of X that belongs to each surface is:

$\displaystyle \dfrac{h_i}{\sum^i h_i}X=x_i$

but $x_i=h_iC_iPT_i$ then

$\displaystyle \dfrac{X}{\sum^i h_i}\dfrac{1}{PT_i}X=C_i$

It seems something is missing, because the answer only depends on PT and not $h_i$. Any ideas?

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