A botanical research station analyzes 50 plant samples for the presence of three specific enzymes: Alpha (AAA), Beta (BBB), and Gamma (ΓΓΓ). The distribution of these enzymes across the samples is summarized in the following data:
\begin{itemize} \item 9 samples contain only enzyme A.\item6samplescontainbothenzymes. \item 6 samples contain both enzymes .\item6samplescontainbothenzymesA,andandandB,butnot, but not ,butnotΓ.\item11samplescontainonlyenzyme. \item 11 samples contain only enzyme .\item11samplescontainonlyenzymeB.\item4samplescontainbothenzymes. \item 4 samples contain both enzymes .\item4samplescontainbothenzymesB,andandandΓ,butnot, but not ,butnotA.\item8samplescontainonlyenzyme. \item 8 samples contain only enzyme .\item8samplescontainonlyenzymeΓ.\item12samplescontainnoneoftheseenzymes.\itemNosamplescontainboth. \item 12 samples contain none of these enzymes. \item No samples contain both .\item12samplescontainnoneoftheseenzymes.\itemNosamplescontainbothA,andandandΓ, simultaneously. \end{itemize}
One sample is selected at random.
Show that the probability that the sample contains more than one enzyme is 0.2.
Determine the probability that the sample contains enzyme A A\,A or enzyme BBB (or both).
State the probability that the sample contains both enzyme A A\,A and enzyme ΓΓΓ.
Given that the sample is found to contain at least one of the three enzymes,
calculate the probability that it contains enzyme ΓΓΓ.
Determine, with a reason, whether the presence of enzyme B B\,B and the presence of enzyme Γ Γ\,Γ in a sample are statistically independent events.
