Problema Solution
Dr. Hahn has developed a new procedure that he believes can correct a life-threatening medical condition. If the success rate for this procedure is 82% and the procedure is tried on 12 patients, what is the probability that at least 9 of them will show improvement?
Answer provided by our tutors
P(at least 9 will show improvement) = P(9 will show improvement) + P(10 will show improvement) + P(11 will show improvement) + P(12 will show improvement)
For calculating P(9 will show improvement) we have:
n = 12
r = 9
p = 0.82
q = 1 - p = 1 - 0.82 = 0.18
P(9 will show improvement) = 12C9 * 0.82^9 * 0.18^(12 - 9)
P(9 will show improvement) = 12!/(9!(12 - 9)!) * 0.82^9 * 0.18^3
P(9 will show improvement) = ((10*11*12)/6)* 0.82^9 * 0.18^3
P(9 will show improvement) = 0.215062588
The same way we find the other probabilities:
P(10 will show improvement) = 12C10 * 0.82^10 * 0.18^(12 - 10) = ((11*12)/2)* 0.82^10 * 0.18^2 = 0.29391887
P(11 will show improvement) = 12C11 * 0.82^11 * 0.18^(12 - 11) = 12 * 0.82^11 * 0.18 = 0.243447953
P(12 will show improvement) = 12C12 * 0.82^12 * 0.18^(12 - 12) = 0.82^12 = 0.092420056
We add them all together:
P(at least 9 will show improvement) = 0.215062588 + 0.29391887 + 0.243447953 + 0.092420056
P(at least 9 will show improvement) = 0.844849467
The probability that at least 9 of them will show improvement is 0.844849467.