Replacement
|
Mentor: Let us start by defining our game. We'll consider a bag of marbles. In this bag there are two different colors of marbles, red and blue. In our example we'll say that there are 4 red and 4 blue. We will take two marbles out of the bag without looking inside, and when we look at the first one we find that it is red. What are the chances, theoretically speaking, that the second one is also red? Student 1: Well, that's pretty easy. We know that there are 4 of each color marble and 8 in the bag, so there is a 4/8 chance that we'll draw a red -- hey that's 1/2! -- and another 1/2 chance that we'll draw another red. So the probability of both happening is 1/2 x 1/2 = 1/4. Student 2: But wait a minute. If we draw 1 red marble out doesn't that mean that there are just 7 marbles left in the bag, only 3 of which are red marbles? Wouldn't this make our new probability be 1/2 * 3/7 = 3/14? Mentor: We have two different answers! Maybe the problem is not as simple as it at first appeared to be. Let us look at several easier problems and then return to The Marble Bag later.
Mentor: This whole question hinges on a term called replacement. Both of you are exactly right, but it depends on the way you view the problem. Let's look at an easier problem. Let's say that we have just 4 marbles in the bag, 2 red and 2 blue. The likelihood of drawing a red is 1/2. Now for a moment let's say we didn't put the marble that we just drew, back into the bag. So how many total marbles do we have after pulling the first red out? Student 2: There are 3 total marbles in the bag if you don't put the red one back. Mentor: What is the probability of drawing another red if we don't replace the one we've already drawn? Student 1: Then it would be 1/2*1/3 which is 1/6 wouldn't it? Student 2: That would be because there are only three marbles left. So my answer to the first question was correct then? Mentor: Yes if you choose to look at it that way. Let's go back to our simpler problem of 4 marbles. What if we replace the marble that we have just drawn. We've already established that the probability of drawing one red is 1/2. Now what is the probability of drawing another red if we choose to replace the marble that we have just drawn? Student 1: Then it would be like I said in my answer to the first problem. The probability of drawing another red marble would be 1/2, just like the first time we drew. So the probability of drawing two red ones would be 1/2*1/2 or 1/4. Mentor: Right, it all depends on whether or not we replace the marble or item that we have just withdrawn from the bag. Everytime we have a probability question like this -- choosing multiple objects out of the same set -- we have to decide whether we have replacement or not. This process of choosing objects like marbles out of containers like bags to determine what the contents of the bag look like is called sampling, so we have been talking about sampling with and without replacement! |
