S5 2003 1 Edexcel

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For brevity, write the event of being infected with strain A as A and the event of fully recovering within a week by R. We aim to find:

\mathrm P(A \mid R)

By Bayes’ theorem we have:

\displaystyle \mathrm P(A \mid R) = \frac {\mathrm P(R \mid A) \mathrm P(A)} {\mathrm P(R)}

We know that the probability of fully recovering in a week with strain A is 0.8, so \mathrm P(R \mid A) = 0.8. The probability of having strain A is 0.4, so \mathrm P(A) = 0.4.

We know that 80\% of people with strain A fully recovered within a week, (so a proportion of 0.8 \times 0.4 of all people were infected with strain A and made a full recovery in a week) 60\% for B and 90\% for C. Since these are the only three strains of influenza, (and they are all mutually exclusive, we assume) the probability of making a full recovery within one week is:

\begin{align*}\mathrm P(R \mid A) \mathrm P(A) + \mathrm P(R \mid B) \mathrm P(B) + \mathrm P(R \mid C) \mathrm P(C) & = 0.4 \times 0.8 + 0.1 \times 0.6 + 0.5 \times 0.9 \\ & = 0.83\end{align*}

(this is the law of total probability)

So:

\displaystyle \mathrm P(A \mid R) = \frac {0.8 \times 0.4} {0.83} = \frac {32} {83}