| 1. A jar has 10 marbles, either black or white. 2 marbles are randomly chosen
from the jar. If q is the probability that both will be black, is q > 1/3?
1) Less than 1/2 of the marbles in the jar are white.
2) The probability that 1 white marble and 1 black marble will be chosen together
is 7/15.
A) Statement (1) BY ITSELF is sufficient to answer the question, but statement
(2) by itself is not.
B) Statement (2) BY ITSELF is sufficient to answer the question, but statement
(1) by itself is not.
C) Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question,
even though NEITHER statement BY ITSELF is sufficient.
D) Either statement BY ITSELF is sufficient to answer the question.
E) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question,
meaning that further information would be needed to answer the question.
Solution
This is a dependent probability problem. If you want to find the probability
of choosing 2 black marbles, you will need to figure out the probability that
the first marble will be black and that the second marble will be black. In
this case, the question wants to know if that probability is larger than 1/3.
Statement 1 tells us that less than half the marbles are white, which means
that more than half the marbles are black. The best way to approach this is
to systematically (but quickly) figure out what the probability of two black
marbles is for each scenario. We can do it easily by drawing a chart:
| Black Marbles |
White Marbles |
P(2 Black) |
| 6 |
4 |
6/10 × 5/9 = 1/3 |
| 7 |
3 |
7/10 × 6/9 = 21/45 |
| 8 |
2 |
8/10 × 7/9 = 28/45 |
| 9 |
1 |
9/10 × 8/9 = 4/5 |
| 10 |
0 |
10/10 × 9/9 = 1 |
As you can see, when less than half the marbles are white, the probability of
choosing 2 black marbles can be higher or equal to 1/3, depending on how many
black marbles there are. This is not sufficient.
Statement 2 tells us that the probability of choosing one
black marble and one white marble is 7/15. This is a trap. Since the probability
given is exact, it may seem that only one scenario of black marbles and white
marbles will work. If you work through all the scenarios, you will see that
when there are 7 black marbles and 3 white marbles, the probability of choosing
one of each is 7/15. However, it would also be true in reverse: If there were
7 white marbles and 3 black marbles, the probability would also be 7/15. Therefore,
this is not enough information.
Combining them does give us enough information.
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