# 2 construals "of 100 patients presenting with a lump like the claimant’s in Gregg v Scott, 42 will be ‘cured’ if they are treated immediately."

Are there official terms for these 2 different interpretations of the same statistic?

Lord Hoffmann and Baroness Hale advanced the following arguments against awarding damages for the pure loss of a chance of being cured:

(1) The

Hotson^{82}problem. In cases such asGregg v Scott, it may well be that the claimant never had a chance of being cured, and if this is the case it would be inappropriate to award him damages on the basis that his doctor’s negligence deprived him of a chance of being ‘cured’.^{83}The statisticians tell us that of 100 patients presenting with a lump like the claimant’s inOne way of interpreting this statistic is to say that each of those 100 patients has a 42 per cent chance of being ‘cured’. If this is right, then it would be correct to say that the claimant inGregg v Scott, 42 will be ‘cured’ if they are treated immediately. [emphasis mine]Gregg v Scotthad a 42 per cent chance of being ‘cured’ when he saw the defendant doctor. But there is a different way of reading the statistics. It may be that of 100 patients presenting with a lump like the claimant’s inGregg v Scott, the varying genetic make-ups of the 100 patients mean that 42 of them are certain to be ‘cured’ so long as the lump is treated immediately, and 58 of them have no chance of being ‘cured’ no matter how much treatment they receive. If this is right, then it was more likely than not that the claimant inGregg v Scotthadno chanceof being ‘cured’ and it would therefore be inappropriate to award him damages on the basis that he did have a chance of being cured.

^{82}See above, § 8.6.

^{83}Gregg, at [79]–[81] (per Lord Hoffmann).

N.J. McBride and R. Bagshaw, *Tort Law*, 6th edn (2018), page 285.

## 2 answers

One can try to make the arguments mathematically more precise.

- The first claim is that each patient has a 42 % chance of being cured with the treatment.
- The second claim is saying there is a genetic factor that 42 % of people have, such that the conditional probability of being cured given the factor is one, while the conditional probability without the factor is zero.

The second claim assumes we know more about the disease and probably assumes that there is some way of finding out the presence of this genetic factor, but the extent to which this latter assumption is made depends on the context.

Compare with a similar argument with a coin flip: given a good enough physical model, and given that we have sufficient knowledge of the circumstances, we can calculate the result of the coin flip. (Assume for the sake of argument this to be true.) But if just want to decide which of us takes the trash out and flip a coin about it, this is entirely irrelevant, since we do not in fact have the knowledge or the computational tools to figure out which way the coin flip will land.

As to which is better, this depends on the information and resources we have and the general context. I am not familiar with common law and interpretations of probabilities and dubious claims about the treatability of diseases there.

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The author of this passage is proposing a very simple *hidden variable model*. This model has two variables:

- $A$ — whether the patient has certain unknown genes (this variable is hidden)
- $B$ — whether the treatment succeeds (this variable is observed)

And the model proposes that $A$ influences $B$. Mathematically, this model predicts that $P(B) = P(B|A)P(A)$. The author observes that an observed value of $P(B)$ (42%) is insufficient to infer either $P(B|A)$ or $P(A)$ by themselves, assuming this model.

The two examples the author gives correspond to assuming that $P(A)$ is 1 (everyone's genes are good enough for the treatment—in which case $P(B|A) = P(B) = 0.42$) and that $P(B|A)$ is 1 (the treatment is always effective if the right genes are present—in which case $P(A) = P(B) = 0.42$). If your hidden variable model contains probabilities that identically equal 1, you have an opportunity to eliminate variables from the model (both of these examples represent opportunities to reduce the number of variables from 2 to 1). This process might be called dimensionality reduction, though that usually refers to lossy simplifications of models from more to fewer variables, and eliminating identical variables is lossless.

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