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?
We let $x$ denote the number of slips with a 2 written on them.  (This is the usual tactic of letting a variable denote what we're trying to solve for in the problem.)  Then there are $12-x$ slips with a 7 on them. The probability of drawing a 2 is $\frac{x}{12}$ and the probability of drawing a 7 is $\frac{12-x}{12}$, so the expected value of the number drawn is $$ E = \frac{x}{12}(2) + \frac{12-x}{12}(7) = \frac{84-5x}{12}. $$But we are given that $E=3.25$, so we have the equation $$ 3.25 = \frac{84-5x}{12}. $$This simplifies to $39 = 84 - 5x$, which means that $x = 9$. Thus $\boxed{9}$ of the 12 slips have a 2 written on them.
9
Level 3
Counting & Probability
Problem: 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?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>Let x be the number of slips with a 2, and y be the number of slips with a 7. The total number of slips is given by the equation x + y = 12. The expected value of a slip drawn from the bag can be calculated using the formula for expected value: E(X) = (sum of values * their probabilities). 

The expected value is given as 3.25. Therefore, we can express this as:

E(X) = (2x + 7y) / (x + y) = 3.25.

Substituting y with (12 - x) from the first equation into the expected value equation gives:

(2x + 7(12 - x)) / 12 = 3.25.

This simplifies to:

(2x + 84 - 7x) / 12 = 3.25.

Combining like terms results in:

(84 - 5x) / 12 = 3.25.

Multiplying both sides by 12 to eliminate the denominator yields:

84 - 5x = 39.

Rearranging this equation leads to:

5x = 84 - 39,
5x = 45.

Dividing both sides by 5 gives:

x = 9.

Thus, the number of slips with a 2 is 9.<|END_OF_LATENT|> The final answer is $9$.