The first statement tells us the proportion of red marbles to yellow, and is helpful, but it tells us nothing about the number, relative or absolute, of blue marbles, so it alone does not answer the question; for similar reasons, neither does the second statement alone.

Together, however, they make the picture complete. If there are  yellow marbles, then by Statements 1 and 2, there are, respectively,  red marbles and  blue marbles - and, therefore,  marbles total. The probability of drawing a blue marble is .

We need to know the number of marbles that are yellow, and the number of marbles there are total. If we let  be the number of red, blue, and yellow marbles, respectively, the two statements say that  and . 

These two statements together do not give us enough information. For example, these two situations both fit the conditions given:

Situation 1: , making the probability of drawing a yellow marble 

Situation 2: , making the probability of drawing a yellow marble 

You need to know two things to answer this question - the rank of the added card, and the rank of the removed card. The second statement is useful but not sufficient; the first is irrelevant to the question.

There are two composite numbers on each die, 4 and 6. On one die, the probability of rolling a 4 or a 6 is  ; the probabililty of rolling two composite numbers is .

While experiments such as repeated tossings can give an idea of the probability that a coin will come up heads or tails, they do not provide a definitive answer, so neither statement is helpful here.

The two statements are actually equivalent; if the loaded die comes up any of 2, 3, 4, 5, or 6 with probability , then it will come up 1 with probability , which is exactly the same assertion as Statement 1 makes.

A sum of 2 can only come up one way: a pair of ones. If you know either statement, then you can calculate the probability of tossing a 2 by multiplying , the probability of a die coming up a 1 on the fair die, by , the probability of a 1 coming up on the loaded die.

The only way the sum of the two dice can come up 12 is a pair of sixes, so the only probabilities that count are those of each die coming up 6; Statement 2 is therefore irrelevant to the problem. If you are given Statement 1, however, then you can calculate the probability of tossing a 12 by multiplying:

Box 1 will contain  white marbles; Box 2 will contain  white marbles.

The probability of choosing Box 1 is , and the probability of choosing a white marble from this box is , making the probability of choosing Box 1, then a white marble, .

The probability of choosing Box 2 is , and the probability of choosing a white marble from this box is , making the probability of choosing Box 2, then a white marble, .

The overall probability of choosing a white marble is the sum of these probabilities:

Since both boxes contain the same number of marbles, the one with the most red marbles offers us the greater probability of drawing a red marble. For simplicity's sake, assume that each box contains thirty marbles - these arguments will be valid whatever the number.

From Statement 1, it follows that two-thirds of sixty, or forty, marbles total are red, but either box can contain more than the other.

From Statement 2, it follows that fifteen marbles in Box 2 are red, but no information about Box 1 follows.

From both statements, however, it follows that there are fifteen red marbles in Box 2 and twenty-five red marbles in Box 1, so Box 1 has more red marbles and offers the greater chance of drawing a red marble.

If only Statement 1 is assumed, there is no way of knowing which box offers the highest probability of drawing a red marble, since there is no way to tell which box has the higher portion of its marbles red without knowing how their numbers of white marbles or their total numbers of marbles compare. A smiliar argument holds for Statement 2 alone.

If both statements are known, however, each box has the same number of red marbles, and Box 2 contains more marbles overall - this makes Box 1 the box with the highest portion of its marbles red, and the box that offers the greater chance of drawing a red marble in a random draw.