A local eye clinic has two doctors. Dr. Kutallot recommends glass 60% of the time and corrective surgery 40% percent of the time. Dr. Lenzie recommends glasses 70% of the time and 30% of the time. 60% of the patients are treated by Dr. Lenzie. No patients are treated by both doctors. If a patient is selected random, what is the probability that surgery is recommended?

18%
34%
35%
58%
70%

K = .40 * .40 = ?

L = .30 *.60 = ?

Either-or probabilities are found by adding the individual probabilities.

To calculate the probability that surgery is recommended, we need to consider the recommendations of both doctors and the proportion of patients treated by each.

Let's denote:
- A: Event that surgery is recommended
- D1: Event that the patient is treated by Dr. Kutallot
- D2: Event that the patient is treated by Dr. Lenzie

According to the given information:
P(A|D1) = 0.4 (Dr. Kutallot recommends surgery 40% of the time)
P(A|D2) = 0.3 (Dr. Lenzie recommends surgery 30% of the time)
P(D1) = 0.4 (40% of the patients are treated by Dr. Kutallot)
P(D2) = 0.6 (60% of the patients are treated by Dr. Lenzie)

We can now calculate the overall probability of surgery recommendation using the law of total probability:

P(A) = P(A|D1) * P(D1) + P(A|D2) * P(D2)
= 0.4 * 0.4 + 0.3 * 0.6
= 0.16 + 0.18
= 0.34

Therefore, the probability that surgery is recommended is 34%.

So the correct answer is 34%.

To solve this problem, we can break it down into a series of smaller steps.

Step 1: Calculate the probability of randomly selecting each doctor.
Since 60% of the patients are treated by Dr. Lenzie, the probability of randomly selecting Dr. Lenzie is 60%. Similarly, since no patients are treated by both doctors, the probability of randomly selecting Dr. Kutallot is 100% - 60% = 40%.

Step 2: Calculate the probability of surgery being recommended by each doctor.
Dr. Kutallot recommends surgery 40% of the time, so the probability of surgery being recommended by Dr. Kutallot is 40%. Similarly, Dr. Lenzie recommends surgery 30% of the time, so the probability of surgery being recommended by Dr. Lenzie is 30%.

Step 3: Use the law of total probability to calculate the overall probability of surgery being recommended.
The law of total probability states that the probability of an event A is equal to the sum of the probabilities of A given each possible condition. In this case, surgery being recommended is the event (A), and the possible conditions are being treated by Dr. Kutallot or being treated by Dr. Lenzie.

P(surgery) = P(surgery | Kutallot) * P(Kutallot) + P(surgery | Lenzie) * P(Lenzie)

P(surgery) = 0.40 * 0.40 + 0.30 * 0.60
P(surgery) = 0.16 + 0.18
P(surgery) = 0.34

Therefore, the probability that surgery is recommended is 34%, which corresponds to the answer choice "34%".