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### Drawing Marbles

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Date: 02/01/97 at 18:53:26
From: Kayla
Subject: probablity

A jar contains two red marbles, three blue marbles, and four green
marbles.  Niki draws one marble from the jar, and then Tom draws a
marble from those remaining.  What is the probablity that Niki draws a
green marble and Tom draws a blue marble?  Express your answer as a
common fraction.
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Date: 02/03/97 at 21:28:42
From: Doctor Wallace
Subject: Re: probablity

Hi Kayla!

The probability of some event happening is expressed as a fraction or
a decimal from 0 to 1.  A probability of 0 means the event can never
happen.  A probability of 1 means the event is certain to happen.

One useful rule is that to find a basic probability, with all outcomes
equally likely, we make a fraction like this:

number of chances of our event
---------------------------------
number of total chances

For example, suppose we have a jar with 4 red marbles and 6 blue.  We
want to find the probability of drawing a red one at random.  So our
event is "drawing a red marble."  The probability of this is:

number of red marbles    (the chances of our event)
-----------------------
total marbles in jar     (the number of total chances)

In our example, this is 4/10 which is 2/5, reduced.  So the
probability of drawing a red marble is 2/5.  This is because all the
outcomes are equally likely.  That is, any individual marble has the
same chance of being drawn.  If we numbered all the marbles, what is
the probability of picking out no. 5?  Well, there is only 1 number 5
marble, and still 10 marbles in the jar, so the answer is 1/10.

Now suppose we have 2 events.  Let's say that Niki is going to draw 1
marble, and then Tom is going to draw one from the remaining marbles.
What is the probability that Niki gets a blue one?  What is the
probability that Tom gets a red one?

Again, we use our fraction.  When Niki draws, there are 10 marbles in
the jar, of which 6 are blue, so her probability of drawing a blue is
6/10 or 3/5.  After she draws, it is Tom's turn.  But now there are
only 9 marbles left.  4 of these are red, so his probability of
drawing a red marble is 4/9.

Now, it is important to distinguish in any probability problem how
many events you have.  Here we have figured the probability for TWO
events.  The first is that Niki draws a blue marble.  The second is
that Tom draws a red one AFTER Niki has drawn.

But, suppose we want to know the probability of the ONE event: "Niki
draws a blue marble AND Tom draws a red one."  It seems like the same
question, but it isn't.  The reason is that now we have more than one
way this could happen.  We could have:

(1) Niki draws a blue, then Tom draws a blue
(2) Niki draws a blue, then Tom draws a red
(3) Niki draws a red, then Tom draws a blue
(4) Niki draws a red, then Tom draws a red

These are the only 4 possibilities.  They are not all equally likely,
however.  When we have ONE event which is made up of two separate
events with the word AND, we multiply the individual probabilities to

So, the probability of (2) above, is:

Niki draws a blue = (3/5)  times   Tom draws a red = (4/9)

which is 12/45 or 4/15.

How about the probability of (1)?

Well, we already figured the probability of Niki drawing blue; it's
3/5.  How about Tom drawing blue also?  Well, after Niki draws blue,
there are 9 marbles left, and 5 blue, so its 5/9.  And so 3/5 times
5/9 is 3/9 or 1/3.

See how to do it?

You should be able to do your problem, now.  I got a bit lengthy here,
since I can't tell from your question if it's meant to be just 1 event
with an "and."  I think that it is.  But, if not, you can also figure
out just the individual probabilities for the two marble draws, as
well.

If you have any more questions, don't hesitate to write back!

-Doctor Wallace,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
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Associated Topics:
High School Probability
Middle School Probability

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