How to evaluate Binomial(X,Y)?

Question

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For part c) of this probability question, I have never seen this kind of notation before and was wondering how the answer was evaluated? I understand that more than half of a group of 5 would be 3 and above, but how was 0.0127 found?

Answer 1

In part a), it is found that $p= 0.115$ is the probability that a chosen 25 year old male is over 6 foot 2. Part c) asks for the probability involving 5 males. So the solution lets $X$ be a random variable for a binomial distribution with $n=5$ and $p=0.115$. Then $P(X\geq 3)$ is found using the distribution, which can be calculated as $P(X\geq 3) = {5\choose3}p^3(1-p)^2 + {5\choose4}p^4(1-p)+{5\choose5}p^5 = 0.0127$.