A fair, six-sided die is rolled eight times, to form an eight-digit number. What is the probability that the resulting number is a multiple of 8? Express your answer as a common fraction.

Total number of numbers using 1-6

= 6^8 = 1679616

For any number to be divisible by 8, its last 3 digits must be divisible by 8
but these last two digits must contain only the digits from 1 to 6, with no zeros.

possible cases are:
112 136 144 152
216 224 232 256 264
312 336 344 352
416 424 432 456 464
512 536 544 552
616 624 632 656 664

Hoping that I didn't miss any, I count 27

so the front 5 numbers could be anything
there are 6^5 or 7776 different front numbers, each of those could have 27 different last three numbers
so there are 7776x27 or 209952 which are divisible by 8

prob that a number is a multiple of 8
= 209952/1679616
26244/209952
= 6561/52488
= 1/8

mmmmh, very interesting

That is right

That is indeed correct.

A number is a multiple of 8 if and only if the number formed by its last three digits is a multiple of 8.

We want to find the three-digit multiples of 8, where all the digits are between 1 and 6. (More precisely, we want to find how many there are.) First, we find those where the first digit is 1. These are 112, 136, 144, and 152.

Next, we find those where the first digit is 2. These are 216, 224, 232, 256, and 264.

Next, we find those where the first digit is 3. Since 200 is a multiple of 8, we can find these numbers by adding 200 to all the numbers where the first digit is 1. This gives us 312, 336, 344, and 352.

Next, we find those where the first digit is 4. Since 200 is a multiple of 8, we can find these numbers by adding 200 to all the numbers where the first digit is 2. This gives us 416, 424, 432, 456, and 464.

We can do the same calculation in the cases where the first digit is 5 and 6. This gives us $4 + 5 + 4 + 5 + 4 + 5 = 27$ three-digit multiples of 8.

There are a total of $6^3 = 216$ possible outcomes for the last three digits, all of which are equally probable, so the probability that the number is divisible by 8 is $27/216 = \boxed{1/8}$.

Why did the dice go to therapy?

Because it couldn't stop rolling with itself!

But seriously, let's figure out the probability. To be divisible by 8, the number formed by rolling the six-sided die eight times must be divisible by 8.

There are 6 choices for each roll, so there are a total of \(6^8\) possible ways to roll the die.

We need to find the number of outcomes that give us a multiple of 8.

To do this, we need to consider the last three digits of the eight-digit number. The last three digits must form a multiple of 8.

Out of the six possible numbers for each digit, there are \(2^3 = 8\) possibilities that form a multiple of 8 (e.g., 168, 712, 536).

Therefore, the probability is \(\frac{8}{6^8} = \boxed{\frac{1}{729}}\).

But don't worry, you won't need therapy after figuring out this probability!

To find the probability that the resulting number is a multiple of 8, we need to determine how many possible numbers are multiples of 8 and divide it by the total number of possible outcomes.

A number is divisible by 8 if the three rightmost digits form a multiple of 8. Since a die has six sides, each digit can take values from 1 to 6. However, the last digit cannot be 2 because 2, 12, 22, etc. are not multiples of 8. Thus, the last digit can take values 1, 3, 4, 5, or 6.

The remaining two digits can form any two-digit multiple of 8, from 16 to 96. As such, there are 10 possible numbers: 16, 24, 32, 40, 48, 56, 64, 72, 80, and 96.

Therefore, there are 5 options for the last digit and 10 options for the other two digits, resulting in a total of 5 * 10 = 50 possible numbers that are multiples of 8.

Since each of the eight dice rolls has 6 possible outcomes, the total number of possible outcomes is 6^8.

The probability is then given by:
Number of favorable outcomes / Total number of possible outcomes
= 50 / (6^8)
= 50 / 1679616

Hence, the probability that the resulting eight-digit number is a multiple of 8 is expressed as the fraction:

50 / 1679616