Of those vaccinated against a particular disease, 70% develop sufficient levels of antibodies to avoid contracting the disease when exposed to it. If 100 people who have been vaccinated are exposed to the disease, what is the probability that at most 80 people develop sufficient levels of antibodies to avoid contracting the disease?
In other words: 70% success, 30% failure, 100 people, find probability of 80 or less succeeding.
I got the correct answer, but it took me way too long to solve this problem. Here is my working out:
Pr(X≤80) = 1 - Pr(X>80)
= 1 - (Pr(X=81) + Pr(X=82) + Pr(X=83) + Pr(X=84) + Pr(X=85) + Pr(X=86) + Pr(X=87) + Pr(X=88) + Pr(X=89) + Pr(X=90) + Pr(X=91) + Pr(X=92) + Pr(X=93) + Pr(X=94) + Pr(X=95) + Pr(X=96) + Pr(X=97) + Pr(X=98) + Pr(X=99) + Pr(X=100))
= 1 - (100c81 * (0.7)^81 * (0.3)^19 + 100c82 * (0.7)^82 * (0.3)^18 + 100c83 * (0.7)^83 * (0.3)^17 + ... + 100c100 * (0.7)^100 * (0.3)^0)
= 0.9911
= 99.11% chance
Except in reality, I couldn't just write the "...", I had to actually extend my answer out completely. It took me about 5-10 minutes just to type it all in to my calculator. Surely there is a quicker way of solving this...
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