Before the addition of the jacks, the deck had 13 clubs and 39 other cards. The 4 jacks from the other deck include 1 club and 3 other cards, so the altered deck contains 14 jacks and 42 other cards. That makes the odds against drawing a club at random 

 - that is, 3 to 1, just as in the unaltered deck

In an unaltered deck, 4 of the 52 cards, or  of the cards, are kings. We are looking for an action that preserves this ratio. 

We see that if the clubs from another deck are added, then there are 5 kings (the 4 original kings plus an added king) and 65 cards total (the original 53 plus the 13 added clubs). 5 of the 65 cards, or  of the cards, are kings, thereby preserving the ratio. 

Of the other choices, none of them change the number of kings in the deck, but all change the total number of cards. Therefore, the ratio of kings to total cards must change. 

Adding the four sevens from another deck to a 53-card deck will yield a 57-card deck with 14 hearts - the original 13 hearts and the added seven of hearts - and 43 other cards. Therefore, the odds against drawing a heart will be 43 to 14.

In an unaltered deck, 13 of the 52 cards, or  of the cards, are spades. We are looking for an action that preserves this ratio. 

If the red fours and the black fives, including the five of spades, are removed, then there are 12 spades left out of 48 cards, which means that  of the cards are spades. This is the correct choice.

Of the other choices, none of them change the number of spades in the deck, but all change the total number of cards. Therefore, the ratio of spades to total cards must change.

Consider first all of the possible ways the men may be arranged, which is

Now, consider all of the ways that Aaron and Gary could be seated beside each other; it may be easier to visualize by drawing it out:

1. A G _ _
2. G A _ _
3. _ A G _
4. _ G A _
5. _ _ A G
6. _ _ G A

As seen, there are six possibilities.

Finally, for each of these cases, Craige and Boone could be seated in one of two ways.

So the probability that Aaron and Gary will be seated beside each other is:

Out of the 100 tiles, there are nine "A" tiles, twelve "E" tiles, nine "I" tiles, eight "O" tiles, and four "U" tiles. This is a total of 

tiles out of 100. 

This makes the probability of drawing one of these tiles

Eight decks of 52 cards, with 4 jokers added, comprise a total of

cards. A standard deck of 52 cards has 13 spades, so the eight decks together have a total of

spades. This makes the probability of a random draw resulting in a spade

As seen in the diagram below, nine of the twenty-five balls in the box are both gray and marked with odd numbers. 

This makes the probability that a random selection will yield one of these balls  .

Out of the whole numbers from 1 through 9, there are four prime numbers - 2, 3, 5, and 7. As seen in the diagram below, there are five out of twenty-five balls that are both white and marked with a prime number.

This makes the probability of drawing one of these balls .

The undesirable rolls are 5, 6, and 8. As can be seen in the diagram below, out of thirty-six equally probable outcomes possible with the roll of two fair six-sided dice, fourteen are undesirable, and twenty-two are desirable.

The odds against an undesirable roll are 

 - that is, 11 to 7.