What is the probability that at most \\(\\var{n2}\\) of them will develop symptoms?  [[0]]

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The probability of succumbing to a particular virus is \\(\\var{p1}\\).

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The probability that at most \\(\\var{n2}\\) of them will develop symptoms is expressed as \\(P(X\\le \\var{n2})\\)

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The binomial distribution gives: \\(P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}\\)

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\\(P(X=0)=\\binom{\\var{n1}}{0}(\\var{p1})^{0}(\\var{q})^{\\var{n1}}\\)

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\\(P(X=0)=(1)(1)(\\simplify{{q}^{{n1}}})\\)

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\\(P(X=0)=\\var{prob_0}\\)

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Similarly

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\\(P(X=1)=\\binom{\\var{n1}}{1}(\\var{p1})^{1}(\\var{q})^{\\simplify{{n1}-1}}\\)

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\\(P(X=1)=\\var{prob_1}\\)

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And

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\\(P(X=2)=\\binom{\\var{n1}}{2}(\\var{p1})^{2}(\\var{q})^{\\simplify{{n1}-2}}\\)

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\\(P(X=2)=\\var{prob_2}\\)

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So

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\\(P(X\\le 1)=\\var{prob_0}+\\var{prob_1}=\\var{prob_a}\\)

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\\(P(X\\le 2)=\\var{prob_0}+\\var{prob_1}+\\var{prob_2}=\\var{prob_b}\\)

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", "statement": "

The probability of succumbing to a particular virus is \\(\\var{p1}\\).

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A group of \\(\\var{n1}\\) patients were exposed to the virus.

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", "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}]}], "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}