(Solved): 1. Horseshoe crabs play an imporant role in medical research. Their blood contains a protein that c ...

1. Horseshoe crabs play an imporant role in medical research. Their blood contains a protein that causes the blood to clot in the presence of endotoxins. Because of this their blood is used in quality control testing for vaccines, injectable drugs, intravenous solutions, and implantable medical devices as a measure of protecting against using products that have been contaminated with endotoxins. When the crabs are harvested for their blood some of these crabs are then used commercially for food or bait, and others are returned to the ocean. Suppose a study was conducted to investigate the mortality rate of crabs returned to the ocean after blood harvesting. The study found that $25\%$ of crabs returned to the ocean after blood harvesting die within 24 hours. (a) [1 mark] Suppose an independent sample of 40 horseshoe crabs are harvested for blood and returned to the ocean. What is the expected number of crabs that will survive longer than 24 hours after being returned to the ocean? (b) [1 mark] Suppose an independent sample of 32 crabs is harvested for blood and returned to the ocean. What is the probability that exactly 9 of these crabs die within 24 hours of being returned to the ocean? (c) $[2$ marks] Suppose an independent sample of 18 crabs is harvested for blood and returned to the ocean. If it is known that at least 3 crabs die within 24 hours of being returned to the ocean, what is the probability that at least 8 crabs die within 24 hours of being returned to the ocean?