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Actually, there is an error in my previous calculations but there is an error in their calculations too. Here is the exact table of probabilities (used by them, reconstructed by me):

Correct Probability
0 36.79%
1 36.81%
2 18.33%
3 6.25%
4 1.38%
5 0.42%
6 0.00%
7 0.02%
-------
Total 100.00%

Note: There is no chance of getting 6 right because in that case you would actually get all seven right.

The probability of Natasha getting 4 or more exactly right is 1.38 + 0.42 + 0.02 = 1.82% or 1 in 55. But that is if she had 7 medical conditions and 7 people to assign them to. There were only 6 conditions and 7 people (one person had no condition at all). When you redo the calculations with that information you get the following table:

Correct Probability
0 42.09%
1 36.74%
2 15.77%
3 4.37%
4 0.89%
5 0.12%
6 0.02%
-------
Total 100.00%

The probability of Natasha getting 4 or more conditions exactly right is 0.89 + 0.12 + 0.02 = 1.03% or 1 in 97.

In addition to this, there is scientific and statistical method to be considered. First is the 'null hypothesis', which in this case would state that, all things considered, Natasha is no different from any other person. To check this we run a test and we use an alpha level (a cut off point) to excluded the null hypothesis. The usual alpha level (commonly used in normal statistical analysis) is 5%. CSICOPS believed they set their alpha level at 0.44% (which was actually 0.14%) (at least 5 right out of a possible 6) which is extremely harsh in my opinion in either case.

Natasha achieved 1,82% on their table, actually 1.03% on my table (at least 4 right out of 6). That is extremely unlikely by pure chance if she is an ordinary person. The expected number of correct answers for a normal person would be no more than 2. Try it yourself and you will see. Either way it is better than 5%.

The calculations posted previously were erroneous, but the ones here have been double checked and I am very confident in them.

Appendix:

This is the program used to generate the tables in this letter (please consider it public domain):

----------------------------------------------------------------------
#include
#include

int main(int argc, char *argv[]) {
int i;
int j;
int tmp;
int count;
int choice;
int actual[7];
int guess[7];
int result[8];

// Initialise
srand(time(NULL));
for (i = 0; i < 7; i++) {
actual = i + 1;
guess = i + 1;
result = 0;
}
result = 0;
// Play ten million guessing games
for (j = 0; j < 10000000; j++) {
// Randomise conditions (actual)
for (i = 0; i < 7; i++) {
choice = rand()%7;
tmp = actual[choice];
actual[choice] = actual;
actual = tmp;
}
// Randomise conditions (guess)
for (i = 0; i < 7; i++) {
choice = rand()%7;
tmp = guess[choice];
guess[choice] = guess;
guess = tmp;
}
// How many were right guesses?
count = 0;
for (i = 0; i < 6; i++) { // 6 conditions only
if (actual == guess)
count++;
}
// Record results
result[count]++;
}
for (i = 0; i < 8; i++) {
printf("=\t%8d\t%8.8g%%\n", i, result,
((float) result)/100000.0);
}
return 0;
}
----------------------------------------------------------------------