There are 100 patients in a hospital with a certain disease. Of these, 10 are selected to undergo a drug

treatment that increases the percentage cured rate from 50 percent to 75 percent. What is the probability
that the patient received a drug treatment if the patient is known to be cured?

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To calculate the probability that a patient received a drug treatment if the patient is known to be cured, we can use Bayes' theorem.

Let's define the following events:
A: Patient received a drug treatment
B: Patient is cured

We are given that there are 100 patients, and 10 of them received the drug treatment. This means P(A) (probability of receiving the drug treatment) is 10/100, which simplifies to 1/10.

We are also given that the drug treatment increases the cure rate from 50% to 75%. This means P(B|A) (probability of being cured given that the patient received the drug treatment) is 75% or 0.75.

To find the probability that a patient received a drug treatment given that the patient is cured, we need to calculate P(A|B) (probability of receiving the drug treatment given that the patient is cured).

Using Bayes' theorem, the formula would be:

P(A|B) = (P(B|A) * P(A)) / P(B)

where P(B) is the probability of being cured.

Let's plug in the values:

P(A|B) = (0.75 * 1/10) / P(B)

To find P(B), we need to consider both cases:
1. Patients who received the drug treatment and were cured (A and B)
2. Patients who did not receive the drug treatment but were cured (not A and B)

P(B) = P(A and B) + P(not A and B)

Since there are 100 patients and 10 of them received the drug treatment, and the cure rate is 75% for patients who received the treatment, we have:

P(A and B) = (1/10) * 0.75 = 0.075

For patients who did not receive the drug treatment, there are 90 patients, and the cure rate is 50%, so:

P(not A and B) = (9/10) * 0.5 = 0.45

Now we can calculate P(B):

P(B) = P(A and B) + P(not A and B) = 0.075 + 0.45 = 0.525

Substituting this value back into the equation for P(A|B):

P(A|B) = (0.75 * 1/10) / 0.525 = 0.143

Therefore, the probability that the patient received a drug treatment if the patient is known to be cured is 0.143 or 14.3%.

To solve this problem, we can use conditional probability. Let's break it down step by step:

Step 1: Calculate the probability of a patient being cured without receiving the drug treatment:
Since the percentage cured rate without the drug treatment is 50 percent, the probability of a patient being cured without the drug treatment is 0.50.

Step 2: Calculate the probability of a patient receiving the drug treatment:
Out of the 100 patients, only 10 were selected to undergo the drug treatment. Therefore, the probability of a patient receiving the drug treatment is 10/100, which is 0.10.

Step 3: Calculate the probability of being cured given that the patient received the drug treatment:
The percentage cured rate with the drug treatment is 75 percent, which means the probability of being cured given that the patient received the drug treatment is 0.75.

Step 4: Calculate the probability of being cured:
To calculate the probability of being cured, we need to consider both cases: receiving the drug treatment and not receiving the drug treatment.

Probability of being cured = (Probability of being cured without the drug treatment) * (Probability of not receiving the drug treatment) + (Probability of being cured with the drug treatment) * (Probability of receiving the drug treatment)

Probability of being cured = (0.50) * (0.90) + (0.75) * (0.10) [Since the probability of not receiving the drug treatment is 1 - 0.10 = 0.90]

Step 5: Calculate the probability of receiving the drug treatment given that the patient is known to be cured:
We are given that the patient is cured. We want to find the probability that the patient received the drug treatment. This can be calculated using Bayes' theorem.

Probability of receiving the drug treatment given that the patient is cured = (Probability of being cured with the drug treatment) * (Probability of receiving the drug treatment) / (Probability of being cured)

Probability of receiving the drug treatment given that the patient is cured = (0.75) * (0.10) / [(0.50) * (0.90) + (0.75) * (0.10)]

By evaluating this expression, we can find the probability that the patient received a drug treatment if the patient is known to be cured.