First, let's find the proportion of patients arriving in each stage: $P(\text{Stage 1}) = 0.10$ $P(\text{Stage 2}) = 0.40$ $P(\text{Stage 3}) = 0.30$ $P(\text{Stage 4}) = 1 - (0.10 + 0.40 + 0.30) = 0.20$

Next, we calculate the survival rates for each stage: $P(\text{Survive} | \text{Stage 1}) = 1 - 0.10 = 0.90$ $P(\text{Survive} | \text{Stage 2}) = 1 - 0.20 = 0.80$ $P(\text{Survive} | \text{Stage 3}) = 1 - 0.30 = 0.70$ $P(\text{Survive} | \text{Stage 4}) = 1 - 0.50 = 0.50$

The joint probabilities of being in a stage and surviving are: $P(\text{Stage 1} \cap \text{Survive}) = 0.10 \times 0.90 = 0.09$ $P(\text{Stage 2} \cap \text{Survive}) = 0.40 \times 0.80 = 0.32$ $P(\text{Stage 3} \cap \text{Survive}) = 0.30 \times 0.70 = 0.21$ $P(\text{Stage 4} \cap \text{Survive}) = 0.20 \times 0.50 = 0.10$

Total probability of survival $P(\text{Survive}) = 0.09 + 0.32 + 0.21 + 0.10 = 0.72$

Using Bayes' theorem, the probability that the patient was in stage 4 given they survived is: $P(\text{Stage 4} | \text{Survive}) = \frac{P(\text{Stage 4} \cap \text{Survive})}{P(\text{Survive})} = \frac{0.10}{0.72} \approx 0.13888$ or $13.88%$.