Addendum

The following letter, from S. D. Turner, contains some surprising information:

Your bit about the two black and two red cards reminds me of an exercise I did years ago, which might be called iV-Card Monte. A few cards, half red, half black, or nearly so, are shown face up by the pitchman, then shuffled and dealt face down. The sucker is induced to bet he can pick two of the same color.

The odds will always be against him. But because the sucker will make erroneous calculations (like the 2/3 and 1/2 in your 2:2 example), or for other reasons, he will bet. The pitchman can make a plausible spiel to aid this: "Now, folks, you don't need to pick two blacks, and you don't need to pick two reds. If you draw either pair you win!"

The probability of getting two of the same color, where there are R reds and B blacks, is:

This yields the figures in the table [see Figure 22], one in lowest-terms fractions, the other in decimal. Only below and to the left of the stairstep line does the sucker get an even break or better. But no pitchman would bother with odds more favorable to the sucker than the 1/3 probability for 2:2, or possibly the 2/5 for 3:3.

Surprisingly, the two top diagonal lines are identical. That is, if you are using equal reds and blacks, odds are not changed if a card is removed before the two are selected! In your example of 2:2, the probability is 1/3 and it is also 1/3 when starting with 2:1 (as is evident because the one card not selected can be any one of the three). The generality of this can be shown thus: If B=R and B=R-1 are substituted into (1), the result in each case is R-1/ 2R-1.

Some readers sent detailed explanations of why the arguments behind the fallacies that I described were wrong, apparently not realizing that these fallacies were intended to be howlers based on the misuse of the principle of indifference.

Figure 22

Red Cards

1

2

3

4

5

6

7

1

1

3

3

1

2

2

2

5

5

3

7

3

3

5

15

7

7

a

11

13

4

4

3

21

28

9

9

5

4

1

7

i

7

7

2

15

11

11

3

11

A

27

31

-Ê.

JL

4

18

45

35

66

13

13

Z

23

31

17

ia

43

7

9

45

55

33

39

91

13 26

2

.333

.333

3

.500

.400

.400

4

.600

.466

.429

.429

5

.667

.524

.465

.444

.444

6

.714

.572

.500

.467

.455

.455

7

.750

.611

.533

.491

.470

.462

.462

8

.778

.645

.564

.515

.488

.472

26 .490

13 26

12 25

13 .480

26 .490

Probability of drawing two cards of the same color

Several readers correctly pointed out that although Pascal did invoke the principle of indifference by referring to a coin flip in his famous wager, the principle is not essential to his argument. Pascal posits an infinite gain for winning a bet in which the loss (granting his assumptions) would always be finite regardless of the odds.

Efron's nontransitive dice aroused almost as much interest among magicians as among mathematicians. It was quickly perceived that the basic idea generalized to k sets of n-sided dice, such as dice in the shapes of regular octahedrons, dodecahedrons, icosahedrons, or cylinders with n flat sides. The game also can be modeled by k sets of n-sided tops, spinners with n numbers on each dial, and packets of n playing cards.

Karl Fulves, in his magic magazine The Pallbearers Review (January 1971) proposed using playing cards to model Efron's dice. He suggested the following four packets: 2, 3, 4, 10, J, Q; 1, 2, 8, 9, 9, 10; 6, 6, 7, 7, 8, 8; and 4, 5, 5, 6, Q, K. Suits are irrelevant. First player selects a packet, shuffles it, and draws a card. Second player does the same with another packet. If the chosen cards have the same value, they are replaced and two more cards drawn. Ace is low, and high card wins. This is based on Efron's third set of dice where the winning probability, if the second player chooses properly, is 11/17. To avoid giving away the cyclic sequence of packets, each could be placed in a container (box, cup, tray, etc.) with the containers secretly marked. Before each play, the containers would be randomly mixed by the first player while the second player turned his back. Containers with numbered balls or counters could of course be substituted for cards.

In the same issue of The Pallbearers Review cited above, Columbia University physicist Shirley Quimby proposed a set of four dice with the following faces:

3, 4, 5, 20, 21, 22 1, 2, 16, 17, 18, 19 10, 11, 12, 13, 14, 15 6, 7, 8, 9, 23, 24

Note that numbers 1 through 24 are used just once each in this elegant arrangement. The dice give the second player a winning probability of 2/3. If modeled with 24 numbered cards, the first player would select one of the four packets, shuffle, then draw a card. The second player would do likewise, and high card wins.

R. C. H. Cheng, writing from Bath University, England, proposed a novel variation using a single die. On each face are numbers 1 through 6, each numeral a different color. Assume that the colors are the rainbow colors red, orange, yellow, green, blue, and purple. The chart below shows how the numerals are colored on each face.

Face Red Orange Yellow Green Blue Purple

The game is played as follows: The first player selects a color, then the second player selects another color. The die is rolled and the person whose color has the highest value wins. It is easy to see from the chart that if the second player picks the adjacent color on the right—the sequence is cyclic, with red to the "right" of purple—the second player wins five out of six times. In other words, the odds are 5 to 1 in his favor!

To avoid giving away the sequence of colors, the second player should occasionally choose the second color to the right, where his winning odds are 4 to 2, or the color third to the right where the odds are even. Perhaps he should even, on rare occasions, take the fourth or fifth color to the right where odds against him are 4 to 2 and 5 to 1 respectively. Mel Stover has suggested putting the numbers and colors on a 6-sided log instead of a cube.

This, too, models nicely with 36 cards, formed in six piles, each bearing a colored numeral. The chart's pattern is obvious, and easily applied to n2 cards, each with numbers 1 through n, and using n different colors. In presenting it as a betting game you should freely display the faces of each packet to show that all six numbers and all six colors are represented. Each packet is shuffled and placed face down. The first player is "generously" allowed first choice of a color, and to select any packet. The color with the highest value in that packet is the winner. In the general case, as Cheng pointed out in his 1971 letter, the second player can always choose a pile that gives him a probability of winning equal to (n-\)/n.

A simpler version of this game uses 16 playing cards. The four packets are:

AS, JH, QC, KD KS, AH, JC, QD QS, KH, AC, JD JS, QH, KC, AD

Ace here is high, and the cyclic sequence of suits is spades, hearts, clubs, diamonds. The second player wins with 3 to 1 odds by choosing the next adjacent suit, and even odds if he goes to the next suit but one.

These betting games are all variants of nontransitive voting paradoxes, about which there is extensive literature.

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