The negation of a proposition is the proposition "Not ," or, "It is not true that ."

The negation of "William Shakespeare is dead" is "It is not true that William Shakespeare is dead," or, simply, "William Shakespeare is alive."

A combination is defined as an unordered subset of a set; a permutation is an ordered subset. The key to answering this question is the phrase " the order...need not be the same." This is an indication that this is a combination.

For any whole numbers , where ,

Setting :

.

This conditional statement has a false antecedent , so, by the principles of logic, it is considered to have truth value "true." The truth value of the consequent is irrelevant.

If a card is drawn at random from a standard deck of 52, the probability of drawing one of the 26 black cards is ; the probability of drawing an ace is . Now, examine each of the three scenarios.

(1) If the ace of spades is replaced with the joker, this leaves 25 black cards and 3 aces out of a total of 52 cards. The probability of drawing one of the 25 black cards is ; the probability of drawing an ace is .

(2) If the joker is added, there are still 26 black cards and 4 aces, but the deck now has 53 cards. The probability of drawing one of the 25 black cards is ; the probability of drawing an ace is .

(3) If the ace of spades is removed, this leaves 25 black cards and 3 aces out of a total of 51 cards. The probability of drawing one of the 25 black cards is ; the probability of drawing an ace is .

In all three cases, both probabilities have changed.

If the odds in favor of an event are to , the probability of that event is . Setting  and , the probability of the event is

.

The odds in favor of an event are equal to the ratio of the probability of the event to that of the opposite event. Therefore, we want to determine :

,

or 17 to 8 in favor.

The fairness or unfairness of the game is a function of the expected value, which can be calculated by multiplying the probability of each outcome by its value, and adding the products.

We will examine the value of the game to Jack; a positive value indicates a gain to Jack, and a negative value indicates a gain to Jill.

Since only ranks matter in this game, there are thirteen equiprobable outcomes. Four are favorable to Jill, and the other nine are favorable to Jack. There are two events, a win for Jill and a win for Jack, which we will call  and , respectively. Their probabilities and their values to Jack are:

: A king, queen, jack, or ten is drawn, and Jill wins.

Probability: 

Value to Jack:

: A card of any of the other nine ranks is drawn, and Jack wins:

Probability: 

Value to Jack: 

The expected value of one play of the game to Jack is

in Jack's favor.

Two cards are drawn from the deck without regard to order, so the sample space  is the set of all combinations of two cards from a set of twelve. The size of this sample space is 

The event is the set of all combinations of two cards from the set of four kings. The size of this event space is

The probability of the event is

The fairness or unfairness of the game is a function of the expected value, which can be calculated by multiplying the probability of each outcome by its value, and adding the products.

We will examine the value of the game to Jack; a positive value indicates a gain to Jack, and a negative value indicates a gain to Jill.

Since only suits matter in this game, there are four equiprobable outcomes. One is favorable to Jill, and the other three are favorable to Jack. There are two events, a win for Jill and a win for Jack, which we will call  and , respectively. Their probabilities and their values to Jack are:

: A spade is drawn, and Jill wins.

Probability: 

Value to Jack: 

: A card of any of the other three suits is drawn, and Jack wins:

Probability: 

Value to Jack: 

The expected value of one play of the game to Jack is

 The game is fair.