There are   possible rolls of two fair six-sided dice, each of which will come up with equal probability. The set of rolls is shown below, with the ways to roll a 7 or less indicated.

There are 21 ways out of 36 to roll a sum of 7 or less, making the probability of this outcome .

Box 1 contains the odd-numbered balls, of which there are

.

One of these balls is marked with a "1", making the probability of drawing this ball .

Box 2 contains the even-numbered balls, of which there are

.

Two of these balls are marked with a "2", making the probability of drawing such a ball .

Between two fractions with the same numerator, the one with the lesser denominator is the greater, so , and the greater probability is that of drawing a "2" from Box 2.

Box 1 contains the odd-numbered balls, of which there are

.

Nine of these balls are marked with a "9", making the probability of drawing one of these balls .

Box 2 contains the even-numbered balls, of which there are

.

Ten of these balls are marked with a "10", making the probability of drawing such a ball .

We can compare these fractions by expressing both with a common denominator. , so we can rewrite these probabilities as 

and 

.

The probability that a ball randomly drawn from Box 1 will be marked "9" is the greater.

Each number will be represented by one red ball, so there will be ten red balls in the box.

The prime numbers - the numbers that have only 1 and themselves as factors - are 2, 3, 5, and 7, so there will be four green balls.

The composite numbers - the numbers that have more than two factors - are 4, 6, 8, 9, and 10, so there will be five blue balls. 

Note that 1 is neither prime nor composite.

Including the unmarked yellow ball, the total number of balls is , five of which are blue, so the probability of drawing a blue ball at random is .

One fourth of the cards in a standard deck of 52, not including the joker, will be diamonds; if two standard decks are shuffled together, then one fourth of those cards will be diamonds. The probability that Mary will draw a diamond is therefore .

However, when the joker is added to John's deck, there are more cards, but just as many diamonds. Therefore, the probability of John drawing a diamond must decrease to slightly less than . This gives Mary a greater probability of drawing a diamond.

As seen from the table below, out of the thirty-six equally probable rolls, eighteen result in the same color facing upward on both dice (indicated by an "S"); eighteen result in different colors facing upward (indicated by a "D").

The events occur with the same frequency, so they also occur with the same probability.

As seen from the table below, out of the thirty-six equally probable rolls, twenty result in an even sum (these are set apart with red squares), and sixteen result in an odd sum.

Since even sums happen with greater frequency, the probability of an even sum is greater.

The diagram below shows how the balls are separated. 

Box A contains 13 balls, of which 4 are white; the probability of a random draw from this box resulting in the draw of a white ball is . 

Box B contains 12 balls, of which 6 are white;  the probability of a random draw from this box resulting in the draw of a white ball is .

Rewriting the probabilities in terms of a common denominator:

, so , making the draw of a white ball more likely from Box B.

The rolls that yield a product greater than or equal to 20 are:

These are 8 out of 36 rolls, so the probability of getting one of them is 

The gray region is the region outside both  and  and is therefore the set of natural numbers that are multiples of neither 3 nor 5.

1,565 and 3,890 can be eliminated as choices as each is divisible by 5: the former ends in 5 and the latter ends in 0.

7,431 and 3,966 can be eliminated as each is divisible by 3:  and . 

7,916 is not divisible by 5 as it ends in 6; it is not divisible by 3 as . Therefore, it is the correct choice.