Question: With each purchase at SlurpeeShack, you receive one random piece of the puzzle seen at right. Once you collect all 12 pieces, you get a free Slurpee! What is the expected value for the number of purchases you will need to make in order to collect all 12 pieces?
My solution: The probability to collect any piece is p=1/(12).
The expected value to collect any piece i at a step is E(i)=1/p= 12 steps.
By linearity of expectation: the expected value of the sum of random variables is equal to the sum of their individual expected values, regardless of whether they are independent.
So, why can't we write E(x)= E(1) + E(2)+ E(3)+....E(12)=144.
The answer is 37.
Can anyone explain where am I wrong in my approach.
Note that $P(\text{Get a new piece}|\text{Have x number of pieces})= \frac{12-x}{12}$. Hence the expected number of attempts when you have $x$ pieces is $\frac{1}{P}=\frac{12}{12-x}$. Summing these up, we get
$$\sum_{n=0}^{11} \frac{12}{12-n}=\sum_{n=1}^{12} \frac{12}{n} =\frac{12}{12}+\frac{12}{11}+\dotsb+\frac{12}{2}+\frac{12}1 =\frac{83,711}{2,310}=37.2385.$$ So 37, (I'd personally round up to 38, but whatever) is the expected number of purchases.