"The spontaneous remission rate of all cancers, lumped together, is
estimated to be something between one in ten thousand and one in a
hundred thousand. If no more than 5 percent of those who come to
Lourdes were there to treat their cancers, there should have been
something between 50 and 500 'miraculous' cures of cancer alone.
Since only three of the attested 65 cures [accepted by the R C Church
as miraculous cures] are of cancer, the rate of spontaneous remission
a Lourdes seems to be lower than if the victims had just stayed at
—From Carl Sagan, The Demon-Haunted World p. 221.
I will reconstruct the references independently because the book has a very poor references section.
- spontaneous remission of cancer can be estimated between 1 and 10 cases every million (so it's much rarer than Sagan assumed) (source)
- it is estimated that 200 million people have visited Lourdes since 1860 (source)
- there are 67 recognised "miracle healings" at Lourdes, of which only 5 are cancer-related (source)
- cancer accounts for way more than 5% of deaths, so we can assume it's an underestimation. From here we can verify that at least 16.8% of male deaths and 11.7% of female death are due to cancer, at least in UK.
There is no particular preference for cancer victims to go to Lourdes over victims of other illnesses, so we can estimate that at most 12-17% of the critically/terminally ill are there for cancer-related reasons (one of the preconditions for a "miraculous cure" is a diagnosed disease).
Not all people go to Lourdes for a terminal illness though, so the more conservative 5% figure that Sagan provided potentially compensates for this.
In other words, if 5% of people coming to Lourdes are there to cure cancer, then their number would amount to 10 million people.
Out of a set of 10 million cancer victims, we should normally expect between 10 and 100 cases of spontaneous remission of the disease, statistically speaking.
However, only 5 cases of "cancer miracles" are reported from Lourdes, making the healing powers of Lourdes statistically insignificant over pure chance.
Most of the other alleged "cures" are pre-1970 and related to TBC and MS, which are now either curable or known to have high rates of temporary remission. Funnily enough, the number of alleged "cures" has diminished as medical science has learned to heal patients of the most common ailments and to properly diagnose common remissions.