It’s a typical Saturday night. You’re at your favorite pub, contemplating the true meaning of infinite cardinalities, when a burly-looking biker plops down on the stool next to you. Just as you are about to get your mind around p(p(R)), biker dude slaps three strange-looking dice on the bar and challenges you to a $100 wager. His rules are simple. Each player selects one die and rolls it once. The player with the lower value pays the other player $100. Naturally, you are skeptical, especially after you see that these are not ordinary dice. Each die has the usual six sides, but opposite sides have the same number on

them, and the numbers on the dice are different, as shown here;

Biker dude notices your hesitation, so he sweetens his offer: he will pay you $105 if you roll the higher number, but you only need pay him $100 if he rolls higher, and he will let you pick a die first, after which he will pick one of the other two. The sweetened deal sounds persuasive since it gives you a chance to pick what you think is the best die, so you decide you will play.

But which of the dice should you choose?

Let us practice the 4 steps modelling method;

### 1. Die A versus Die B

- Step 1: Find the sample space.
- Step 2: Define events of interest.
- Step 3: Determine outcome probabilities.
- Step 4: Compute event probabilities.

### 1. Die A versus Die B:

### Show all above steps and find What is the probability of A winning against B?

### 2. Die B versus Die C :

### Show all steps and find What is the probability of C winning against B?

### 3. Die C versus Die A:

### Show all steps and find What is the probability of A winning against C.

### 4. Show that A>B>C>A meaning: A beats B beats C beats A.

5. Rolling Twice: Show that

## A**
**

If each player rolls twice, the tree diagram will have four levels and 3^4= 81 outcomes. This means that it will take a while to write down the entire tree diagram. But it’s easy to write down the first two levels and then notice that the remaining two levels consist of nine identical copies of the tree. The probability of each outcome is (1/3)^4 =1/81 and so, once again, we have a uniform probability space. This means that the probability that A wins is the number of outcomes where A beats B divided by 81.

To compute the number of outcomes where A beats B, we observe that the two rolls of die A result in nine equally likely outcomes in a sample space SA in which ….

Repeat the above for two role problem in which the sum of the two roles is used to find the winner.

### 6. Show that with two roles each turn; the strength of die reverses that is:

A**How can it be that A is more likely than B to win with one roll, but B is more likely to win with two rolls? Well, why not? The only reason we’d think otherwise is our unreliable, untrained intuition.**

Biker Dude and Non-linear Orders

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It’s a typical Saturday night. You’re at your favorite pub, contemplating the true meaning of infinite cardinalities, when a burly-looking biker plops down on the stool next to you. Just as you

are about to get your mind around p(p(R)), biker dude slaps three strange-looking dice on the bar and challenges you to a $100 wager. His rules are simple. Each player selects one die and

rolls it once. The player with the lower value pays the other player $100. Naturally, you are skeptical, especially after you see that these are not ordinary dice. Each die has the usual six

sides, but opposite sides have the same number on

them, and the numbers on the dice are different, as shown here;

B

Biker dude notices your hesitation, so he sweetens his offer: he will pay you $105 if you roll the higher number, but you only need pay him $ 100 if he rolls higher, and he will let you pick a

die first, after which he will pick one of the other two. The sweetened deal sounds persuasive since it gives you a chance to pick what you think is the best die, so you decide you will play.

But which of the dice should you choose?

Let us practice the 4 steps modelling method;

1. Die A versus Die B

• Step 1: Find the sample space.

1. Die A versus Die B

• Step 1: Find the sample space.

• Step 2: Define events of interest.

• Step 3: Determine outcome probabilities.

• Step 4: Compute event probabilities.

die A

die B

winner

probability

of outcome

1/3

A

1/9

1/9

5 1/3

• B

1/3

B

2 1/3

1/9

A

1/9

6 1/3

1/3

1

5 1/3

9

1/3

A

1/9

B

1/9

1/3

A

7 1/3

1/9

1

5

1/3

1/3

A

1/9

7 1/3

1/3

А

1/9

9

1/3

B

1/9

1. Die A versus Die B:

Show all above steps and find What is the probability of A winning against B?

2. Die B versus Die C:

Show all steps and find What is the probability of C winning against B?

3. Die C versus Die A:

Show all steps and find What is the probability of A winning against C.

4. Show that A>B>C>A meaning: A beats B beats C beats A.

5. Rolling Twice: Show that A 10

8

2

6

6

5

12

10

13

14

10

14

14

18

To compute the number of outcomes where A beats B, we observe that the two rolls of die Aresult in nine equally likely outcomes in a sample space SA in which ….

Repeat the above for two role problem in which the sum of the two roles is used to find the winner.

6. Show that with two roles each turn; the strength of die reverses that is:

A