${A}$, ${B}$ and ${C}$ are taking part in a game in which a prize is hidden in one of $n$ id

Question

${A}$, ${B}$ and ${C}$ are taking part in a game in which a prize is hidden in one of $n$ identical closed boxes, where $n > 4$. First, ${A}$ opens at random two boxes in an attempt to find the prize. If ${A}$ fails to find the prize, ${B}$ then opens at random two of the remaining boxes in an attempt to find the prize. If ${B}$ fails to find the prize, ${C}$ opens all the remaining boxes and finds the prize. Find, in terms of $n$, the probability that ${A}$ finds the prize.[1] Show that the probability that B finds the prize is $\frac{2}{n}$.[1] Find the range of values of $n$ for which the probability that ${C}$ finds the prize is more than $8$ times the probability that either ${A}$ or ${B}$ find the prize.[3] The events $Q$ and $R$ are such that ${P}\left( Q\cap R \right)=x$, $\mathrm{P}\left( Q\cap {R}' \right)=0.1$, ${P}\left( {Q}'\cap R \right)=\mathrm{P}\left( Q \right)$ and ${P}\left( \left( Q\cup R \right)' \right)=\mathrm{P}\left( Q \right)+2\mathrm{P}\left( R \right)$. By using a Venn diagram, or otherwise, find the exact value of $x$.[3]

Accepted Answer

(a)(i) $\frac2n$ (ii) $\frac2n$ (iii) $n>36$ (b) $x=\frac1{14}$

N2024 P2 Q7

N2024 P2 Q7

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