There is a problem that goes as follows: Suppose that every month in the year I buy a lottery ticket and that each month I win a prize with probability 1/4. Let X be the number of months that I win a prize. What is the value of E[X]?
Thus far, I've worked out that : $$P(X=k) = {12 \choose k}\cdot\frac{3}{16}^k$$ From this, I believe that E[X] should be:
$$E[X] = \sum_{k=1}^{12} k\cdot{12 \choose k}\frac{3}{16}^k$$
However, the result gives me a result such as 14.89, which makes no sense in the context of the problem. How can one win 14 months in the range of only 12 months? Could someone point out what is wrong with my equation?
As requested in the comments:
The problem with your sum is that the probabilities are wrong. As this is a binomial process, we have $$P(X=k)=\binom {12}k \times \left(\frac 14\right)^k\times \left(\frac 34\right)^{12-k}$$
When you sum these terms, you get $E=3$ which is much more plausible.
As a simpler alternative to computing this sum, you could just work with indicator variables and invoke the Linearity of Expectation. Thus, we let $X_i$ be the indicator variable for month $i$ (so $X_i=1$ if you win that month, and $X_i=0$ if you don't). Then, of course $E[X_i]=\frac 14$ so $$E[X]=E\left[ \sum X_i\right]=\sum E[X_i]=12\times \frac 14 = 3$$