Probability of breast cancer I'm having trouble with a probability problem I've been trying to solve for a while. It's about the accuracy of breast cancer testing. Relevant probabilities are listed below, where:

*

*"$\text{cancer}$" is the event "has breast cancer".

*"$+$" is the event "tests positive for breast cancer".

$P(\text{cancer}) = \frac{12}{1000}$
$P(+|\text{cancer}) = \frac{11}{12}$
$P(+) = \frac{31}{1000}$
$P(\text{cancer}|+) = 0.355$
This last line is a result from a previous problem. The next part involves updating the probability of having cancer, but I'm having trouble figuring out what the answer is.
In the next part of the question, there's a woman who has tested positive and her doctor says she's part of a population for which there's a 40% chance of breast cancer.
I need to find the probability that the woman has cancer.
I am confused by this update to the cancer probability, but i will assume that this means $P(\text{cancer})$ has changed.
I also assume this means I need to find a new value for $P(\text{cancer}|+)$, but I'm not getting this right.
$P(+ | \text{cancer}) = \frac{11}{12} = \frac{P(\text{cancer} | +)\cdot P(+)}{P(\text{cancer})} = \frac{P(\text{cancer} | +) \cdot \frac{31}{1000}}{0.40}$
$P(\text{cancer} | +) = \frac{11}{12} \cdot 0.40 \cdot \frac{1000}{31} = 11.828$
The result can't be correct because it's way over 1.
How can i fix this? Thank you in advance for any insight.

By Bayes' Theorem:
$$P(cancer|+) = \frac{P(+|cancer) P(cancer)}{P(+)}$$
Where $$P(+) = P(+|cancer)P(cancer)+ P(+|no-cancer)P(no-cancer)$$
$$P(+|cancer) = \frac{11}{12}$$
$$P(cancer) = \frac{4}{10}$$
Therefore,
$$P(cancer|+) = {\frac{11}{12} \cdot \frac{4}{10} \over \frac{11}{12} \cdot \frac{4}{10} + \frac{6}{10} \cdot P(+|no-cancer)}$$
To find the rate of false positives for the test, P(+|no-cancer), we can use the information from the general population, that $P(+)= \frac{31}{1000}$ and that $P(cancer) = \frac{12}{1000}$.
Then, $$ \frac{31}{1000} = \frac{11}{12} \cdot \frac{12}{1000} + P(+|no-cancer) \cdot \frac{988}{1000} $$
Rearrange to get $$ P(+|no-cancer) = \frac{5}{247}$$
Plug this back into the previous equation to get $$P(cancer|+) = \frac{2717}{2807} \approx 0.968$$
