Question: A bag contains 6 red marbles,9 blue marbles,5 green marbles. You withdraw one marble,replace it,and then withdraw another marble. What is the probability that you do not pick two green marbles?

You are drawing out two marbles independently of one another (that is, what marble you draw the first time has no impact on what you draw the next time). In fact, you have two identical actions: you are drawing out one marble each time from the same set of possibilities.

So in this case, it is a lot like rolling a standard six-sided die twice. What you roll the first time doesn't matter to the second roll. The chance of rolling a "4" on a single roll is 1 in 6 because only 1 side out of 6 possible sides has a "4" on it. The chance of NOT rolling a "4" is 5 out of 6.

Now if you roll the same die again, your chance of rolling a "4" is 1 out of 6 for that roll as well.

There is something in math we call the "Fundamental Counting Principle" that says that when we have several independent choices to make, we multiply them together to figure out the number of possible combinations.

So we can figure out the chance of NOT rolling two "4"s by multiplying the possible choices: 6 x 6 = 36 choices. But only one of these pairs choices has two "4"s, so the chance of rolling two "4"s is 1 in 36. The chance of something NOT happening is always the complement of the chance it will happen: that's 35 in 36.

You have to be a bit careful: we can multiply the chance of getting a "4" on each roll: 1/6 x 1/6 = 1/36, then take the complement: 35/36. That's the correct way to do it. Many students make the mistake of saying the chance of NOT getting a "4" on each roll is 5/6, so they assume then can multiply 5/6 x 5/6 to get 25/36, which is clearly a different - and wrong - answer.

Here's why it is wrong: Let's say you roll a 2 on your first roll and a 4 on your second roll. Clearly, that is a "miss" because we didn't roll two fours, but it is not included in the 5/6 x 5/6 because we didn't account for the possibility that one or the other can be a 4 as long as both are not 4s.

With your question, you just have one extra wrinkle: there are 5 green marbles out of 20. You can either work with 5/20 (in place of the 1/6 in our example with the dice) or better yet, recognize that 5/20 is just 1/4 and then use that in place of 1/6.

I hope this helps!

Stephen La Rocque>