Math Geniuses needed!?

Question

Imagine you roll two idintical 12-sided dice (regular dodecahedrons). You roll them at the same time, and make a fraction out of the two numbers that come up. To make the fraction, use the smaller number as the numerator and the larger number as the denominator. Obviously, if the dice show the same number, it doesn't matter which is the numerator (the fraction will be equal to 1). What is the probablility that the resulting fraction is less than 1/2?

Answer 1

Let's change the probability problem into a counting problem by using a more classic definition of probability:

p = number of favorable outcomes
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total number of outcomes

There are a total of 144 different outcomes. How many of them are favorable?

If we roll a 1 on 1 die, we can roll 3, 4, 5, 6, 7, 8, 9, 10, 11, or 12 on the other die.

If we roll a 2 on 1 die, we can roll 5, 6, 7, 8, 9, 10, 11, or 12 on the other die.

If we roll 3 on 1 die, we can roll 7, 8, 9, 10, 11, or 12 on the other die. We could also roll 1, but that has already been counted.

If we roll 4 on 1 die, we can roll 9, 10, 11, or 12 on the other die. 1 has already been counted.

If we roll 5 on 1 die, we can roll 11 or 12 on the other die. 1 and 2 have already been counted.

Nothing that we haven't already counted will work if 6 is rolled on one die. The same is true for all numbers greater than 6.

So far, we have 30 favorable outcomes. But there is nothing special about one die that doesn't apply to the other die.

Suppose that one die is red and the other is blue. Rolling a 1 on the red die and a 3 on the blue die gives 1/3 - a favorable outcome listed above. But rolling a one on the blue die and a 1 on the red die also gives 1/3 by the rules of the problem, and this is just as favorable as the previous occurrence, even though both of them are separate outcomes.

Given that all of the outcomes listed above involve distinct numbers, we can double that number.

It appears as though 60 outcomes are favorable of the 144 total outcomes.

This probablity is therefore 60/144, or 5/12.

If someone sees a flaw in my logic, please share!

Answer 2

I am assuming the sides are numbered 1 through 12.

About the only way I see to do this is by counting carefully. Fortunately, we don't have to count too high... :)

If the lower number is 1, the higher number must be at least 3 (10 possible ways)

If the lower number is 2, the higher number must be at least 5 (8 possible ways)

If the lower number is 3, the higher number must be at least 7 (6 possible ways)

If the lower number is 4, the higher number must be at least 9 (4 possible ways)

If the lower number is 5, the higher number must be at least 11 (2 possible ways)

Total: 30 ways. Double this number, since either die can have the lower number. So, 60 rolls can produce the roll you want.

Now, the total number of possible rolls is 12 * 12 = 144. Therefore, the chance of the dice roll you want is 60/144 = 41.66 %

Answer 3

Possibillities
Roll
a 1 and any number greater than 2
Probability = 2 * 1/12 * 10/12 = 5/36

a 2 and any number greater than 4
Probability = 2 * 1/12 * 8/12 = 4/36

a 3 and any number greater than 6
Probability = 2 * 1/12 * 6/12 = 3/36

a 4 and any number greater than 8
Probability = 2 * 1/12 * 4/12 = 2/36

a 5 and any number greater than 10
Probability = 2 * 1/12 * 2/12 = 1/36

So required probability = (1 + 2 + 3 + 4 + 5)/36
= 15/36
= 5/12

Answer 4

The answer should be 5/12. There are 144 total possibilities (12 x 12). Of these, 12 will give you an answer of 1, and 12 more will give you an answer of 1/2 (1-2, 2-1, 2-4, 4-2, etc). Of the remaining possibilities (there are 120 of them), half (60) should be less than 1/2, and half should be greater than 1/2.

60/144 = 5/12.

Answer 5

Easy enough to do with a spreadsheet. There are 60 fractions less than 0.5 out of the 144 possible. Probability is therefore 0.4167.

If you were to say 'less than or equal to 0.5', there are 72 which qualify, so the probability is 0.5.

Answer 6

I think it's 1 to 12, but I'm not sure, I think I did something like this, is this from data management?

Answer 7

i have no idea what you are talking about, you should put a math grade or age group in parenthesese after the question so that it will be alot easier for other people looking for questions