Collection of medicinal plants

A pensioners' club organised a collection of medicinal plants. Two pensioners could not participate in the collection for health reasons, the others decided to collect dead-nettle, elder, and coltsfoot. All three herbs were collected by 7 pensioners, dead-nettle and also elder by 15 pensioners, dead-nettle and also coltsfoot by 12 pensioners. Coltsfoot was collected by 21, elder by 23, as was dead-nettle. Elder or coltsfoot was collected by 31 pensioners. Determine: What percentage of the club's pensioners took part in the collection action? How many pensioners collected elder and also coltsfoot? What is the probability that a randomly selected pensioner collected coltsfoot and at the same time did not collect dead-nettle?

Final Answer:

p₁ =  94.4444 %
B =  13
p₂ =  0.4286

Step-by-step explanation:

$$\begin{gathered} 8+7+a+b=23 \\ 5+7+b+c=21 \\ 8+7+a+b+5+c=31 \\ a+b=8 \\ b+c=9 \\ a+b+c=11 \\ Row3-Row1\to Row3 \\ c=3 \\ b=9-c=9-3=6 \\ a=8-b=8-6=2 \\ a=2 \\ b=6 \\ n=3+5+8+7+a+b+c=3+5+8+7+2+6+3=34 \\ {p}_1=100\cdot \frac{n}{n+2}=100\cdot \frac{34}{34+2}=94.4444\% \end{gathered}$$

$B=7+b=7+6=13$

$p_2=\frac{b+c}{21}=\frac{6+3}{21}=0.4286$

Try another example