Moderate -0.8 This is a straightforward S1 question testing basic probability concepts: completing a two-way table using row/column totals (simple arithmetic), finding P(J∪W) - P(J∩W), checking mutual exclusivity (P(J∩W)≠0), checking independence (P(J∩W)≠P(J)P(W)), and calculating probabilities with independent events using complement and combinations. All techniques are routine recall with minimal problem-solving required, making it easier than average.
Emma visits her local supermarket every Thursday to do her weekly shopping.
The event that she buys orange juice is denoted by \(J\), and the event that she buys bottled water is denoted by \(W\). At each visit, Emma may buy neither, or one, or both of these items.
Complete the table of probabilities, printed below, for these events, where \(J ^ { \prime }\) and \(W ^ { \prime }\) denote the events 'not \(J\) ' and 'not \(W ^ { \prime }\) respectively.
Hence, or otherwise, find the probability that, on any given Thursday, Emma buys either orange juice or bottled water but not both.
Show that:
(A) the events \(J\) and \(W\) are not mutually exclusive;
(B) the events \(J\) and \(W\) are not independent.
Rhys visits the supermarket every Saturday to do his weekly shopping. Items that he may buy are milk, cheese and yogurt.
The probability, \(\mathrm { P } ( M )\), that he buys milk on any given Saturday is 0.85 .
The probability, \(\mathrm { P } ( C )\), that he buys cheese on any given Saturday is 0.60 .
The probability, \(\mathrm { P } ( Y )\), that he buys yogurt on any given Saturday is 0.55 .
The events \(M , C\) and \(Y\) may be assumed to be independent.
Calculate the probability that, on any given Saturday, Rhys buys:
5
\begin{enumerate}[label=(\alph*)]
\item Emma visits her local supermarket every Thursday to do her weekly shopping.
The event that she buys orange juice is denoted by $J$, and the event that she buys bottled water is denoted by $W$. At each visit, Emma may buy neither, or one, or both of these items.
\begin{enumerate}[label=(\roman*)]
\item Complete the table of probabilities, printed below, for these events, where $J ^ { \prime }$ and $W ^ { \prime }$ denote the events 'not $J$ ' and 'not $W ^ { \prime }$ respectively.
\item Hence, or otherwise, find the probability that, on any given Thursday, Emma buys either orange juice or bottled water but not both.
\item Show that:\\
(A) the events $J$ and $W$ are not mutually exclusive;\\
(B) the events $J$ and $W$ are not independent.
\end{enumerate}\item Rhys visits the supermarket every Saturday to do his weekly shopping. Items that he may buy are milk, cheese and yogurt.
The probability, $\mathrm { P } ( M )$, that he buys milk on any given Saturday is 0.85 .\\
The probability, $\mathrm { P } ( C )$, that he buys cheese on any given Saturday is 0.60 .\\
The probability, $\mathrm { P } ( Y )$, that he buys yogurt on any given Saturday is 0.55 .\\
The events $M , C$ and $Y$ may be assumed to be independent.
Calculate the probability that, on any given Saturday, Rhys buys:
\begin{enumerate}[label=(\roman*)]
\item none of the 3 items;
\item exactly 2 of the 3 items.
\begin{center}
\begin{tabular}{ | c | c | c | c | }
\cline { 2 - 4 }
\multicolumn{1}{c|}{} & $\boldsymbol { J }$ & $\boldsymbol { J } ^ { \prime }$ & Total \\
\hline
$\boldsymbol { W }$ & & & 0.65 \\
\hline
$\boldsymbol { W } ^ { \prime }$ & 0.15 & & \\
\hline
Total & & 0.30 & 1.00 \\
\hline
\end{tabular}
\end{center}
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA S1 2011 Q5 [13]}}