To solve this problem, we can use Bayes' theorem and organize the information given into probabilities:
- Let T be the event that a patient receives the treatment.
- Let I be the event that a patient experiences significant improvement.
From the problem statement, we have:
- P(I|T) = 0.30
- P(I|T^c) = 0.20
- P(T|I) = 0.60
We need to find P(T|I) , the probability that a patient received treatment given that they improved.
Using Bayes' theorem, we can rewrite P(T|I) as:
P(T|I) = \frac{P(I|T)P(T)}{P(I)}
Let's find P(I) using the law of total probability:
P(I) = P(I|T)P(T) + P(I|T^c)P(T^c)
Assume the proportion of patients who received treatment is p , then:
P(T) = p \quad \text{and} \quad P(T^c) = 1 - p
Substitute these into the equation:
P(I) = 0.30p + 0.20(1 - p)
Solve for p using the equation 0.60 = \frac{0.30p}{0.10p + 0.20} . The solution for p is 0.5, meaning that 50% of the patients received treatment.
Now, find P(I) :
P(I) = 0.10p + 0.20 = 0.10(0.5) + 0.20 = 0.25
Finally, the probability that a patient who received the treatment will experience significant improvement is P(I|T) = 0.30 .
Therefore, the probability that a patient who received the treatment will experience significant improvement is **30%**.
\boxed{P(I|T) = 0.30}