Binomial
", "Geometric
", "Poisson
", "Other
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If it's not known whether the equipment is contaminated, what is the probability that $\\var{m}$ cultures survive longer than a week?
\nProbability = ? (to 3 decimal places).
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\nProbability=? (to 3 decimal places).
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a) The binomial distribution.
\nb) Using the law of total probability:
\n\\[\\begin{eqnarray*}\\operatorname{Pr}(\\var{m}\\;\\text{ survive})&=&\\operatorname{Pr}(\\var{m}\\; \\text{survive|uncontaminated})\\operatorname{Pr}(\\text{uncontaminated})+\\operatorname{Pr}(\\var{m} \\;\\text{survive|contaminated})\\operatorname{Pr}(\\text{contaminated})\\\\&=&{\\var{n} \\choose\\var{m}}\\var{p1}^{\\var{m}}(1-\\var{p1})^{\\var{n-m}} \\times (1-\\var{q})+{\\var{n} \\choose\\var{m}}\\var{p2}^{\\var{m}}(1-\\var{p2})^{\\var{n-m}} \\times \\var{q}\\\\&=&\\var{psurvive}\\\\&=&\\var{anspsurvive}\\end{eqnarray*}\\] to 3 decimal places.
\nc) By Baye's theorem:
\n\\[\\begin{eqnarray*}\\operatorname{Pr}\\text{contaminated|}\\var{m}\\;\\text{survive})&=&\\frac{\\operatorname{Pr}(\\var{m} \\;\\text{survive|contaminated})\\operatorname{Pr}(\\text{contaminated})}{\\operatorname{Pr}(\\var{m}\\;\\text{ survive})}\\\\&=&\\frac{{\\var{n} \\choose\\var{m}}\\var{p2}^{\\var{m}}(1-\\var{p2})^{\\var{n-m}} \\times \\var{q}}{\\var{psurvive}}\\\\&=&\\var{anspcont}\\end{eqnarray*}\\] to 3 decimal places.
\n", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}