Considering Box A,
Total number of pens = 3 + 5 = 8 pens
Probability of picking a purple (P) and black (B) pen is given below as,
[tex]\begin{gathered} P(P)=\frac{3}{8} \\ P(B)=\frac{5}{8} \end{gathered}[/tex]Considering Box B,
Total number of pens = 15 + 5 = 20 pens
Probability of picking a purple and black pen is given below as,
[tex]\begin{gathered} P(P)=\frac{15}{20} \\ P(B)=\frac{5}{20} \end{gathered}[/tex]For event 1, probability of choosing a red (R) pen from Box B is zero because there is no red pen in the Box.
Event 1 P(R) = 0
For event 2, probability of choosing a purple or black pen from Box A is,
[tex]P(P\text{ or B)=}\frac{3}{8}+\frac{5}{8}=\frac{3+5}{8}=\frac{8}{8}=1[/tex]Event 2 P(P or B) = 1
For event 3, probability of choosing a purple pen from Box A is,
[tex]P(P)=\frac{3}{8}[/tex]Event 3 (P) = 3/8
For event 4, probability of choosing a black pen from Box B is given below as,
[tex]P(B)=\frac{5}{20}=\frac{1}{4}[/tex]Event 4 P(B) = 1/4
Arranging each events from the least likely to the most likely is in the order below
[tex]\text{Event 1, Event 4, Event 3, Event 2}[/tex]Answer deduced above.