Numerical Problems on Application of Biotechnology In Medicine for JEE and CBSE Exams:

1. Drug Efficacy in Clinical Trials

Problem: A pharmaceutical company is developing a new drug to treat a rare genetic disease. The drug works by targeting a specific gene mutation that causes the disease. In clinical trials, the drug is found to be effective in 70% of patients. If 1000 patients are treated with the drug, how many are expected to benefit?
Solution:
Number of patients expected to benefit = 1000 x 70/100 = 700 patients.

Problem: A biotechnology company is developing a new vaccine to prevent a deadly virus. The vaccine is tested on 500 volunteers and found to be 90% effective in preventing infection. If the vaccine is made available to the public, how many people would need to be vaccinated to prevent 100,000 infections?
Solution:
Number of people needed to be vaccinated = 100,000 / (90/100) = 111,111 people.

Problem: A scientist is studying the effect of a new drug on cancer cells. In vitro experiments show that the drug is able to inhibit the growth of cancer cells by 50%. If the drug is administered to a patient with cancer, what percentage of cancer cells would be expected to be inhibited?
Solution:
Percentage of cancer cells expected to be inhibited = 50%.

Problem: A doctor is monitoring the progress of a patient who has undergone a stem cell transplant. The patient’s immune system is gradually recovering, and the number of white blood cells in the patient’s blood is increasing steadily. The doctor estimates that the patient’s immune system will be fully restored in 6 months. If the patient’s white blood cell count is currently 1000 cells/mL, and it increases by 10% each week, what will be the patient’s white blood cell count in 6 months?
Solution:
After 6 months (24 weeks), the white blood cell count will increase by 240% (10% x 24).
Therefore, the patient’s white blood cell count will be 1000 cells/mL x 240/100 = 2400 cells/mL.

Problem: A genetic counselor is advising a couple who is planning to have children. The woman carries a recessive gene for a rare genetic disorder, and the man is unaffected. The counselor explains that the risk of their children inheriting the disorder is 25%. If the couple decides to have 4 children, what is the probability that none of them will inherit the disorder?
Solution:
Probability that each child will not inherit the disorder = 0.75 (assuming independent assortment of genes).
Probability that none of the 4 children will inherit the disorder = (0.75)^4 = 0.3164.