### Instruction: Suppose I have a bag with 12 slips of paper in it.  Some of the slips have a 2 on them, and the rest have a 7 on them.  If the expected value of the number shown on a slip randomly drawn from the bag is $3.25$, then how many slips have a 2?### Response: To solve this problem, I need to recall the definition of expected value for a discrete random variable.
Expected value is the weighted average of the possible outcomes, where the weights are the probabilities of each outcome.
In this case, the possible outcomes are 2 and 7, and the probabilities depend on how many slips have each number.
Let $x$ be the number of slips that have a 2.
Then there are $12 - x$ slips that have a 7.
The probability of drawing a 2 is $x/12$, and the probability of drawing a 7 is $(12 - x)/12$.
Therefore, the expected value of the number drawn is $(2)(x/12) + (7)(12 - x)/12$.
This expression must equal $3.25$, according to the given information.
I can simplify this equation by multiplying both sides by 12 and expanding the parentheses.
I get $2x + 7(12 - x) = 39$.
Simplifying further, I get $2x + 84 - 7x = 39$.
Combining like terms, I get $-5x = -45$.
Dividing by -5, I get $x = 9$.
This means that there are 9 slips that have a 2.