Expert: Josh - 3/17/2010

Question
Hi, thanks so much, but you never actually answered my question.  Can you plse stick to this exact question, and not give other examples.
A jar has seven yellow marbles, six blue marbles, ten red marbles, and nine green marbles.  One marble is selected at random and then not put back.  Then a second marble is selected at random and not put back.  Finally, a third marble is selected and not put back.  Find the probability that the marbles chosen are blue, red, and green, in that order

Many thanks!

Answer
Hi Bella,

I came across your question in the question pool, without knowing what sort of response you got the first time around.

Generally speaking, for questions which involve "selection without replacement" and "the order matters", the easiest way to work out the probability is by drawing some pictures.

Let Y=yellow, B=blue, R=red and G=green.

Initially, we have 7Y, 6B, 10R and 9G, i.e., 32 marbles in total.
(Hopefully, you can visualize things from this symbolic representation)
I could have drawn Y,Y,Y,Y,Y,Y,Y,B,B,B,B,B,B,...etc. But you get the idea.

For the 1st selection, the probability of getting a "B" is P("B")=6/32; simply because there are 6 blue ones amongst 32 marbles.

Since the Blue marble will not be replaced, we have 7Y, 5B, 10R and 9G, i.e., 31 marbles in total for the next round.

For the 2nd selection, the probability of getting a "R" given that a "B" has already been selected is P("R"|"B")=10/31.

| A quick note on the notation:
| - The vertical bar in P("R"|"B") literally represents the word "given", i.e.,
|   the event of interest is the selection of "R", conditioned upon "B"
|   having previously been selected.
| - Similarly, P("G"|"BR") is the probability of selecting G "given" that the
|   sequence consists of "B" followed by "R" up til this point.
| - If we were to make a 4th selection and say we wanted a "Y" this time, then,
|   we would calculate P("Y"|"BRG"). The event of interest is the selection
|   of a yellow marble, having already selected "B", "R" and "G" strictly in this order.

Continuing from where we left off....

Again, the Red marble is removed without replacement. So, the new situation is depicted by 7Y, 5B, 9R and 9G (with 30 marbles in total).

For the 3rd selection, the probability of getting a "G" given that a "RB" has previously been selected is P("G"|"RB")=9/30.

So, for this chain of events to occur, the overall probability is given by the product of P("G"|"RB"), P("G"|"B") and P("B").

Ans: (6/32)*(10/31)*(9/30).