Step 1: Think of assigning each of the 5 students one activity, and count how many ways match the pattern 3 movie, 1 play, 1 shopping.
Step 2: First pick which 3 of the 5 students go to the movie: $\binom{5}{3} = 10$ ways.
Step 3: From the remaining 2 students, pick which 1 goes to the play: $\binom{2}{1} = 2$ ways. The last student automatically goes shopping.
Step 4: Total number of ways to split 5 students into (3 movie, 1 play, 1 shopping) groups is $10 \times 2 = 20$.
Step 5: Each specific labeled outcome has probability $(0.5)^3(0.3)^1(0.2)^1 = 0.0075$ since choices are independent.
Step 6: Multiply the count of favorable arrangements by the probability of one arrangement:\[ P = 20 \times 0.0075 = 0.15 \]\[\boxed{0.15}\]