**Problem:**

### Solution:

Chance! chance is easy if you know how to do the formula. we know that we have two choices to make. first we’ll pick a jar, and each jar will have a 1/2 chance of being picked. then we’ll pick a marble, and depending how we stack the marbles, we’ll have a (# of red marbles in jar)/(# of total marbles in jar) chance of getting a red one.

for example, say we put all the red marbles into

**JAR A**and all the blue ones into**JAR B**. then our chances for picking a red one are:
1/2 chance we pick

1/2 chance we pick

**JAR A*** 50/50 chance we pick a red marble1/2 chance we pick

**JAR B*** 0/50 chance we pick a red marble
do the math and you get 1/2 chance for a red marble from

**JAR A**and a 0/2 chance for a red marble from**JAR B**. add ‘em up and you get the result = 1/2 chance for picking a red marble.
think about it for awhile and see if you can figure out the right combination. we had a 50/50 (guaranteed) chance in picking a red marble from

**JAR A**, but we didn’t have to have 50 red marbles in there to guarantee those fantastic odds, did we? we could’ve just left 1 red marble in there and the odds are still 1/1. then we can take all those other marbles and throw them in**JAR B**to help the odds out there.
let’s look at those chances:

1/2 we pick

1/2 we pick

**JAR A*** 1/1 we pick a red marble1/2 we pick

**JAR B*** 49/99 we pick a red marble
do the math and add them up to get 1/2 + 49/198 = 148/198, which is almost 3/4.

we can prove these are the best odds in a somewhat non-formal way as follows. our goal is to maximize the odds of picking a red marble. therefore we can subdivide this goal into maximizing the odds of picking a red marble in

**JAR A**and maximizing the odds of picking a red marble in**JAR B**. if we do that, then we will have achieved our goal. it is true that by placing more red marbles into a jar we will increase the chances of picking a red marble. it is also true that by reducing the number of blue marbles in a jar we will increase the odds also. we’ve maximized the odds in**JAR A**since 1/1 is the maximum odds by reducing the number of blue marbles to 0 (the minimum). we’ve also maximized the number of red marbles in**JAR B**. if we added any more red marbles to**JAR B**we would have to take them out of**JAR A**which reduce the odds there to 0 (very bad). if we took any more blue ones out of**JAR B**we would have to put them in**JAR A**which reduce the odds there by 50% (very bad).
it wasn’t really a good proof, but QED anyway :-P

**POST YOUR SOLUTION IF YOU HAVE BETTER SOLUTION**

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