MHB What is the Probability of Grouping All Red Marbles Together?

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The discussion focuses on calculating the probability of grouping all four red marbles together when dividing 36 marbles (32 green and 4 red) into three groups of 12. Participants suggest viewing the problem as a partitioning challenge, where they need to determine the number of ways to arrange the red marbles within the groups. They propose considering different scenarios, such as selecting all red marbles in one group or not selecting them at all. The solution involves calculating the probabilities of these distinct cases and summing them for the final probability. Understanding these cases is essential for solving the probability problem effectively.
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I need help starting the following problem. :confused:

There are (36) marbles in a bag-(32) are green and (4) are red. The marbles are to be divided equally into (3) groups of (12). What is the probability that one of the groups will contain all of the (4) red marbles?
 
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Re: Basic Probability

I would look at this as a partitioning problem. How many ways can we partition 4 objects into 3 groups (2 partitions with 5 potential positions), and how many of those ways have all four objects in one group?
 
Re: Basic Probability

Or another way to think of it is you're "choosing" 12 marbles out of a bag at random. There are a few separate cases you have to consider:

Case #1: Choosing all 4 red marbles out of the bag at once.
Case #2a: Not choosing any red marbles out of the bag, and choosing another 12 marbles, with all 4 of the red marbles in that group.
Case #2b: Not choosing any red marbles out of the bag, then choosing another 12 marbles and not getting any red marbles out of that either.

Find the probabilities of each of these cases and then add them.
 
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