# Probability

## Data Analysis: Probability

Read the following SAT test question and then select the correct answer.

Always read each question carefully and make a note of the bottom line (what you are trying to find).  Assess your options to find the best strategic method and use that method to attack the problem.  When you have an answer, loop back to verify that the answer matches the bottom line.

A jar contains only red marbles and green marbles. If a marble is selected at random from the jar, the probability that a red marble will be selected is $\frac{2}{3}$. If there are 36  green marbles in the jar, how many red marbles are there in the jar?

Bottom Line:  You want to know how many red marbles there are, so use r to represent red and just write r = ?

Assess your Options:  You could try to work backwards from the answer choices to find a number that, when combined with 36, makes the right fraction.  That won’t be any faster than just solving the problem.  Use the probability formula.

Attack the Problem:  The probability formula is:

$\frac{relevant\: outcomes}{total\: outcomes}$

In this problem, you know the red marbles are the relevant outcome, while the red and green marbles together are the total (all that is in the jar).  Use g for the green marbles.  There are 36 green marbles.

$\frac{r}{g + r}=\frac{r}{36+r}$

You have already been given the probability that a red marble will be selected.  Set the formula that you created equal to the probability that you were given.  Then solve for r with cross-multiplication.

$\frac{r}{36+r}=\frac{2}{3}$
3r = 2(36 + r)
3r = 72 + 2r
r = 72

Loop Back:  You solved for your bottom line, so you are ready to look at the answer choices.

(A) 18
(B) 24
(C) 54
(D) 72
(E) 108