Drawing Marbles

Probability Level 4

Evan has a bag with 2 red marbles and 2 blue marbles. He also has a stack of infinite red and blue marbles beside him. Evan draws two marbles at random and puts either a red or blue marble back based on the following conditions:

(A) If the marbles are the same color, Evan puts a marble of the opposite color back in the bag. (ex. if he draws two red marbles, he puts a blue one back from his infinite pile of marbles)

(B) If the marbles are different colors, Evan puts the blue marble back in the bag.

If Evan repeats this process until there is one marble left, the probability that it is red can be written as $\frac{a}{b}$ where $a$ and $b$ are coprime positive integers. Find $a+b$ .

The answer is 13.

1 solution

Geoff Pilling
Jul 21, 2017

Let's work backward.

In order to end up with a red (R), the last two need to be BB (2 blues).

The only way to get BB is if the last three were BBR and you picked BR.

And the only way you can get BBR is if you pulled out BR from the original four (BBRR).

Therefore, in order to end up with one red, the following steps need to be taken:

1) BBRR (pick BR) $\to$ BBR ( $\frac{2}{3}$ probability)
2) BBR (pick BR) $\to$ BB ( $\frac{2}{3}$ probability)
3) BB (pick BB) $\to$ R (100% probability)

So, the total probability is:

$\frac{2}{3} \cdot \frac{2}{3} = \frac{4}{9}$

$4+9 = \boxed{13}$

Drawing Marbles

The answer is 13.

1 solution

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