Question

In a two-player game, five cards, numbered 1 through 5, are placed in a bag. A card is drawn at random, and the players look at the number. Which of the following scoring rules makes a fair game?

If it is greater than 2, Player 1 earns 2 points. If not, Player 2 earns 2 points.

If it is less than 2, Player 1 earns 3 points. If not, Player 2 earns 2 points.

If it is greater than 3, Player 1 earns 2 points. If not, Player 2 earns 2 points.

If it is less than 3, Player 1 earns 3 points. If not, Player 2 earns 2 points.

Answer 1

The last scoring rule makes a fair game

If card value is less than 3, Player 1 earns 3 points. If not, Player 2 earns 2 points.

Explanation:

This is a question which relates to probabilities, complementary probabilities and expected outcomes

Since there are 5 cards in total and only 1 card is picked, the probability of picking a card is the same i.e. 1/5 = 0.2 since each card is identical and therefore has the same likelihood of being drawn

Let's go through each of the rules, find the associated outcomes and probabilities of each of the outcome

The attached table shows a summary of all the possible outcomes of each rule, the probabilities associated with it, the complement of the rule, the complement probability, expected payoffs and who has the advantage for each rule

As an example, take the first row relating to the first rule:
If value of card is greater than 2, Player 1 earns 2 points. If not, Player 2 earns 2 points.

If the value is greater than 2 then the value is either {3, 4 or 5}. There are 3 possible outcomes out of a total of 5 so,

P(card value > 2) = P(3, 4, 5) = 3/5 = 0.6 and Player 1 gets 2 points so the expected value = 0.6 x 2 = 1.2 EP(P1

The complement of this event is

card value <=2 ie card value is either 1 or 2 ie 2 possible out of 5
P(card value <=2) = P(1,2) = 2/5 = 0.4
Note that this probability is 1 - P(card value > 2) = 1- 0.6 = 0.4

If so, Player 2 gets 2 points and his/her expected payoff is 0.4 x 2 = 0.8 EP(P2)

Since EP(P1) > EP(P2) this rule gives player 1 an advantage and therefore not a fair game

The other rows of the table can be interpreted the same way. Essentially we are looking for a rule that gives both players the same expected payoff and therefore each player has the same chance of winning

The table shows that the last rule is the fairest

Answer 2

If it is less than 3, Player 1 earns 3 points.

If not, Player 2 earns 2 points.

Explanation:

Player 1 :

p(N < 3) = p(N = 1 or N = 2) = 2/5

Player 2 :

p(N ≥ 3) = p(N = 3 or N = 4 or N = 5) = 3/5

We notice that :

p(N < 3) × 3 = (2/5) × 3 = 6/5

On the other hand,

p(N ≥ 3) × 2 = (3/5) × 2 = 6/5

since ,the probability player 1 win multiplied by the associated number of points (3)

is equal to

the probability player 2 win multiplied by the associated number of points (2).

Then the game is fair.