Assume that the chances of a person having a skin disease are 40%. Assuming thatskin creams and drinking enough water reduces the risk of skin disease by 30% andprescription of a certain drug reduces its chance by 20%. At a time, a patient canchoose any one of the two options with equal probabilities. It is given that afterpicking one of the options, the patient selected at random has the skin disease. Findthe probability that the patient picked the option of skin creams and drinking enoughwater using the Bayes theorem.Assume E1: The patient uses skin creams and drinks enough water; E2:The patient uses the drug; A: The selected patient has the skin disease

To solve this problem, we will use Bayes' theorem, which is a way of finding a probability when we know certain other probabilities. The formula is:

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

Where:

• P(A|B) is the probability of event A given event B is true.
• P(B|A) is the probability of event B given event A is true.
• P(A) and P(B) are the probabilities of events A and B respectively.

In this case, we want to find the probability that the patient picked the option of skin creams and drinking enough water (E1) given that the patient has the skin disease (A). So we are looking for P(E1|A).

We know that:

• P(A) = 0.4 (the probability of having the skin disease)
• P(E1) = 0.5 (the probability of choosing skin creams and drinking enough water)
• P(E2) = 0.5 (the probability of choosing the drug)
• P(A|E1) = 0.4 * (1 - 0.3) = 0.28 (the probability of having the skin disease given the patient chose skin creams and drinking enough water)
• P(A|E2) = 0.4 * (1 - 0.2) = 0.32 (the probability of having the skin disease given the patient chose the drug)

We also know that P(A) = P(E1) * P(A|E1) + P(E2) * P(A|E2), so we can find P(A) = 0.5 * 0.28 + 0.5 * 0.32 = 0.3.

Now we can use Bayes' theorem to find P(E1|A):

P(E1|A) = P(A|E1) * P(E1) / P(A) = 0.28 * 0.5 / 0.3 = 0.47 (approximately).

So, the probability that the patient picked the option of skin creams and drinking enough water given that the patient has the skin disease is approximately 0.47 or 47%.

To solve this problem, we will use Bayes' theorem, which is a way of finding a probability when we know certain other probabilities. The formula is:

P(E1|A) = P(A|E1) * P(E1) / P(A)

Where:

• P(E1|A) is the probability we are trying to find: the patient used skin creams and drank enough water given that they have the skin disease.
• P(A|E1) is the probability of having the disease given that the patient used skin creams and drank enough water.
• P(E1) is the probability of a patient choosing the option of using skin creams and drinking enough water.
• P(A) is the total probability of having the disease.

From the problem, we know that:

• P(A) = 0.4 (the chance of a person having a skin disease)
• P(E1) = P(E2) = 0.5 (a patient can choose any one of the two options with equal probabilities)
• P(A|E1) = 0.4 * (1 - 0.3) = 0.28 (the risk of skin disease is reduced by 30% if the patient uses skin creams and drinks enough water)
• P(A|E2) = 0.4 * (1 - 0.2) = 0.32 (the risk of skin disease is reduced by 20% if the patient uses the drug)

We also know that P(A) = P(A and E1) + P(A and E2) = P(A|E1)P(E1) + P(A|E2)P(E2) = 0.28 * 0.5 + 0.32 * 0.5 = 0.3

Now we can substitute these values into Bayes' theorem:

P(E1|A) = P(A|E1) * P(E1) / P(A) = 0.28 * 0.5 / 0.3 = 0.47 (approximately)

So, the probability that the patient picked the option of skin creams and drinking enough water given that they have the skin disease is approximately 0.47 or 47%.