Problem of the Week
Problem A and Solution
Which Marble Did I Get?
Problem
James has a collection of marbles. He carries his collection of marbles everywhere he goes in a
bag that you cannot see through. His bag of marbles includes:
• only blue and orange marbles
• only striped and solid marbles
• 15 striped marbles
• 14 solid marbles
• 5 of the striped marbles are blue
• 9 of the solid marbles are orange
You reach into the marble bag and pull out 1 marble.
What is the probability that you will choose a striped, orange, marble?
What is the probability that you will choose a solid, blue, marble?
Solution
First we need to determine the total number of marbles in the bag. We have 15
striped marbles and 14 solid marbles. This means there are 15 + 14 = 29 marbles
altogether.
Second we need to determine how many striped orange marbles are in the bag.
We know that there are 15 striped marbles and 5 of those are blue. This means
that there are 10 remaining striped marbles in the bag that have to be orange.
Finally, since 14 of the marbles are solid, and 9 of the solid marbles are orange,
then the remaining 5 solid marbles must be blue.
Image source: http://www.flickr.com/photos/michaelroper/3584019/
(flikr Creative Commons License)
We can make a chart to describe the situation.
blue orange
striped
5
10
solid
5
9
total
10
19
total
15
14
29
We can use the information in the chart to compute the probabilities.
There would be 10 chances out of 29 that the marble you choose would be striped
and orange, or a probability of 10
29 .
Now let’s compute the probability of choosing a solid, blue marble. Assuming the
two events are independent, we would start with 29 marbles in the bag before
choosing the marble.
Therefore, there would be 5 chances out of 29 that the marble you choose would
5
be solid and blue, or a probability of 29
.
Teacher’s Notes
In Statistics, we need to be very careful to distinguish between independent,
dependent and compound events.
Suppose you wanted to flip a fair coin. Over time, we would expect that this coin
would land on heads half of the time and tails half of the time. In other words, if
we flipped the coin once, the probability that the coin would land on heads is 1
out of 2, or 12 .
Now, suppose you flipped the coin four times in a row, and it landed on heads
each time. What is the probability that if you flip the coin again that you would
land on heads? The probability is still 12 . In this case, the action of flipping the
coin on the fifth try is independent of the previous actions. On that one try, the
coin is just as likely to land on heads as tails.
However, if we changed the question and asked, "If you were to flip a fair coin
five times, what is the probability that all five flips would land on heads?" this is
a completely different problem. The chances of successfully producing the
required output depends on all five of the individual flips. We would consider this
a compound event, where probability depends on the number of possible
outcomes when flipping a coin 5 times in a row. On each flip, there are two
possible outcomes: heads or tails. The total number of combinations of five coin
flips would be: 2 ⇥ 2 ⇥ 2 ⇥ 2 ⇥ 2 = 32. Only one of these combinations would be
all heads. Therefore, the probability of flipping a coin 5 times in a row and seeing
1
heads every time would be 1 out of 32, or 32
.
The marble problem would have been much more difficult if we had asked you to
choose two marbles from the bag and then to figure out the probability that the
first one was orange and striped and that the second one was solid and blue.
These choices would be considered to be dependent events. In this case, the
action of the first choice affects the total number of marbles and possibly the
number of solid, blue marbles in the bag. This dependence will affect the
probabilities.