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1. A jar has 10 marbles, a combination of black and 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:

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. From statement 2 we know that there must be 7 of one color and 3 of the other, and from statement 1 we know that there must be more black than white, so we know there must be 7 black marbles and 3 white marbles.
The correct answer is (C).

2. In a class with 12 children, q of the children are girls. Two children will be randomly chosen simultaneously. What is the value of q?

(1) The probability that two girls will be chosen together is 1/11.
(2) The probability that one boy will be chosen and one girl will be chosen is 16/33.

(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
To answer this question, you can do the math, or you can rely on the experience you have gained thus far. Let’s work out statement 1 by thinking it through:

Statement 1: We know there are a specific number of girls (q). Since each number of girls would yield a different probability of choosing 2 girls, there must be only one specific number that would yield 1/11. So it must be enough information.

Now, statement 2 requires a little more thought. Let’s work it out by doing the math:
Statement 2: This one may seem to follow the same logic, as they are giving us a specific probability. However, this time we are asked to pick one boy and one girl. Look at the following chart to see why this isn’t enough information:

*Note: we will multiply each probability by 2 -- see below (1/12 × 11/11)/2, because we can choose a boy and a girl, or a girl and a boy, and both will yield the desired result.

*Note: we will divide each probability by 2, because we can choose a boy and a girl, or a girl and a boy, and both will yield the desired result.

As you can see, each probability is repeated for inverse combinations of boys and girls. There are two ways to get 16/33, once with 4 boys and 8 girls, and also with 4 girls and 8 boys. This is not enough information. We do not know what q is.
The correct answer is (A).

3. In a hotel with single rooms and double rooms, what is the probability that a room chosen at random will be a double room painted red?

(1) 1/6 of the rooms in the hotel are painted red.
(2) 2/3 of the hotel’s rooms are double rooms.

(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
Statement 1 tells us the fraction of rooms painted red, but we do not learn anything about double rooms.

Statement 2 tells us the fraction of the rooms that are double rooms, but we do not know anything about red rooms.

Putting the statements together still does not give us enough information because we do not know how many of the red rooms are double rooms. Imagine if we said there were 12 rooms. We would then know that 2 were red, and 8 were doubles. But we do not know if any of the doubles are red or not. There is simply no information connecting the two categories, so we cannot solve the probability (E) is the correct answer.

4. Two identical dice are rolled together. If the sum of the dice is 7, what is the probability that one of the numbers showing is a 4?
(A) 1/36
(B) 1/18
(C) 1/6
(D) 11/36
(E) 1/3

Solution
For this problem, we want to calculate the probability of rolling a 4 knowing that the sum of the dice is 7. We therefore do not have to take into account any other combinations of dice other than those that equal 7:

(1,6) (2,5) (3,4) (4,3) (5,2) (6,1)

There are six pairs that equal seven, and 2 of them have a four in them. Therefore, the probability is 2/6 = 1/3.

The correct answer is (E).

5. At 3 pm, Jennifer went into labor. There is a 70% chance her baby will be born each hour that she is in labor. What is the probability that her baby will be born at 6 pm on the same day?
A) .027
B) .063
C) .147
D) .27
E) .343

Solution
This question is an independent probability question in disguise. The probability that her baby will be born each hour does not change. Each hour, there is a 70% chance the baby will be born, which means there is a 30% chance the baby will not be born.

Therefore, we can see that from 3 pm – 4 pm, the baby is not born and the probability of that happening is 30%, from 4 pm – 5 pm, the probability is 30%, and from 5 pm – 6 pm, when the baby is born, the probability is 70%. Since the baby must not be born in the first hour, and must not be born in the second hour, and must be born in the third hour, the probability is (0.3)(0.3)(0.7) = .063
The correct answer is (B).

6. A fair coin is to be flipped four times. What is the probability that the coin will land on the same side on all four flips?
(A) 1/32
(B) 1/16
(C) 1/8
(D) 1/4
(E) 1/2

Solution
The probability the coin will land on the same side is the probability that it will land on all heads or all tails. All heads looks like: H H H H and all tails looks like: T T T T.

The probability of each is 1/2 × 1/2 × 1/2 × 1/2 = 1/16. Since either could happen, we will add the two probabilities together: 1/16 + 1/16 = 2/16 = 1/8.
The correct answer is (C).

The probability, combinations, permutations chapter is complete.

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