Answer:
11.42 boxes
Step-by-step explanation:
For the first box bought, there is a 100% chance of getting a unique toy (since you still don't have any). E₁ = 1.
After that, there is a 4 in 5 chance of getting a unique toy from the next box, the expected number of boxes required is:
[tex]E_2 = (\frac{4}{5})^{-1} = 1.25[/tex]
For the next unique toy, there is now a 3 in 5 chance of getting it:
[tex]E_3 = (\frac{3}{5})^{-1} = 1.67[/tex]
Following that logic, there is a 2 in 5 chance of getting the 4th unique toy:
[tex]E_4 = (\frac{2}{5})^{-1} = 2.5[/tex]
Finally, there is a 1 in 5 chance to get the last unique toy:
[tex]E_5 = (\frac{1}{5})^{-1} = 5[/tex]
The expected number of boxes to obtain a full set is:
[tex]E=E_1+E_2+E_3+E_4+E_5\\E=1+1.25+1.67+2.5+5\\E=11.42\ boxes[/tex]
