# Problem understanding the calculation of normative (Bayesian) base rates

I am having trouble understanding Table 1 of Gigerenzer, Hell, and Blank (1988, PDF, table on page 516): Focusing on the Jack row, it is stated that the mean probabilities of Jack being an engineer were 71.4% in a low base rate condition and 81.3% in a high base rate condition. Thus participants partially took base rates into account. If they were completely ignoring base rates these values would be the same, rather than being separated by 9.9%. So I believe I understand what is going on in the Base Rate Neglect column.

However, I don't understand what is going on in the Bayesian column. There is an explanatory footnote on p516 but to me it was not illuminating. The footnote provides the following equations:

$$p_{70}(E|D)/(1-p_{70}(E|D)) = L\frac{0.7}{0.3}$$ $$p_{30}(E|D)/(1-p_{30}(E|D)) = L \frac{0.3}{0.7}$$

So, since the mean guess for the 70% condition was 81.3% we get

\begin{align} 81.3/18.7 & = L ~ 0.7/0.3 & \\ & = 81.3/18.7 & = L ~2.33 \\ & = 4.35 &= L ~2.33 \\ & = 1.86 &= L \\ \end{align}

However, then when I plug this $L$ value into the $p_{30}$ equation I get a value for $p_{30}(E|D)$ that deviates wildly both from the 'Bayesian' value and from what was actually observed. According to the table the average deviation from the Bayesian should be 16.1%, but according to my calculations it is much higher. It would be greatly appreciated if someone could tell me what I am doing wrong.

### References

• Gigerenzer, G., Hell, W. & Blank, H. (1988). Presentation and content: The use of base rates as a continuous variable.. Journal of Experimental Psychology: Human Perception and Performance, 14, 513. PDF
• This would probably receive more attention on Stats.SE. Aug 6 '12 at 18:07
• @eykanal I think it would be on topic on both sites. Questions at the interface of quantitative methods and psychology/cognitive science are most welcome here as seen by the upvotes. Aug 8 '12 at 8:27

$$p_{30}(E|D) \approx \frac{p_{70}(E|D)}{5.44-4.44p_{70}(E|D)}$$