Question

please help:

one hundred athletes are trying to qualify for the finals. each of them has to pass the preliminary stage and the advanced stage. each athlete passes the preliminary stage with the probability 0.4 and the advanced stage with probability 0.7 independently. each athlete passes stages independently of all other athletes. Let N be the number of people who pass the preliminary stage and fail to pass the advanced stage. estimate the probability P(N ≤ 15)

thank you

Answer 1

The probability that a person passes the preliminary stage and fails to pass the advanced stage is 0.12:

Now take the probability of passing the preliminary stage and failing to pass the final as p:
$$
p=0.12
$$
Otherwise, the person passed the final and passed the preliminary or did not pass the preliminary, q:
$$
q=1-p=0.88
$$
We use a Bernoulli model with parameters n,p,q to find \(P(N\le15)\) :
$$
P(N\le 15)=\sum_{n=0}^{15}{100\choose n}p^nq^{100-n}\approx 0.86\\
$$
$$
\href{http:///www.wolframalpha.com/input/?i=sum+%28100+choose+n%29*%280.12%29%5En+*+%280.88%29%5E%28100-n%29%2C+0%3C%3Dn%3C%3D15}{\text{Click here to see how }}
$$

Does this make sense?