Whooping cough (pertussis) is a highly contagious bacterial infection that was a major cause of childhood deaths before the

Question

Whooping cough (pertussis) is a highly contagious bacterial infection that was a major cause of childhood deaths before the development of vaccines. Approximately 80% of unvaccinated children who are exposed to whooping cough will develop the infection compared to only 5% of vaccinated children.
 In 2007, Bob Jones University ended its fall semester a week early because of a whooping cough outbreak; 158 students were isolated and another 1200 given antibiotics as a precaution. Authorities react strongly to whooping cough outbreaks because the disease is so contagious. Because the effect of childhood vaccination often wears off by late adolescence, treat the Bob Jones students as if they were unvaccinated. It appears that about 1400 students were exposed.
 What is the probability a that at least 75% of these students develop infections if not treated? (Enter your answer rounded to two decimal places.) 
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