Probability of Rolling a Number Less Than 7 on an 8-Sided Die: A Comprehensive Guide

The question of rolling a number less than 7 on an 8-sided die can be understood through a detailed examination of all possible outcomes and their probabilities. Let's break this down step-by-step.

Understanding the Scenario

An 8-sided die (octahedral die) has sides numbered from 1 to 8. When we roll this die, we are interested in the probability that the result is a number less than 7. Such numbers include 1, 2, 3, 4, 5, and 6. Therefore, the favorable outcomes are 6 out of 8 possible outcomes.

Calculating the Probability

The probability of an event is given by the ratio of the number of favorable outcomes to the total number of possible outcomes. For our instance:

Total outcomes: There are 8 possible outcomes when rolling an 8-sided die.

Favorable outcomes: The numbers less than 7 are 1, 2, 3, 4, 5, and 6. This gives us 6 favorable outcomes.

Probability Number of favorable outcomes / Total number of outcomes 6 / 8 0.75

Exploring Exotic Outcomes with Repeated Numbers

Let's explore more complex scenarios where the numbers are repeated. For instance, we might want to roll a die 10 times and determine the probability of rolling a number less than 7. This requires combinatorial analysis.

Combinatorial Analysis for 10 Rolls

When we roll an 8-sided die 10 times, there are (8^{10}) possible outcomes. We need to determine the number of favorable outcomes where the result is always less than 7. This involves a thorough combinatorial approach to count the distinct sequences of 10 rolls where each roll results in a number from 1 to 6.

Let's break down the scenarios:

1. All Numbers from 1 to 6

There are 6 ways each of the 10 rolls can be any of the numbers 1 to 6. This gives us:

Favorable outcomes (6^{10})

2. Numbers with Repeated Occurrences

We need to calculate the number of ways to roll 10 times with repeated numbers from 1 to 6.

a. 4 numbers repeated once: ( binom{6}{4} times frac{10!}{2!^4} )

b. 2 numbers repeated once and one number repeated twice: ( binom{6}{2} times binom{4}{1} times frac{10!}{2!^2 times 3!} )

c. 2 numbers repeated twice: ( binom{6}{2} times frac{10!}{3!^2} )

d. 1 number repeated three times and one number repeated once: ( binom{6}{1} times binom{5}{1} times frac{10!}{3! times 2!} )

e. 1 number repeated four times: ( binom{6}{1} times frac{10!}{5!} )

3. Including Numbers 7 or 8 Once

We need to consider scenarios where the numbers 7 or 8 are included, but the condition still holds that the result is less than 7 in the majority of rolls.

a. 3 numbers repeated once with 7 or 8 included: ( binom{6}{3} times 2 times frac{10!}{2!^3} )

b. 1 number repeated once and 1 number repeated twice with 7 or 8 included: ( binom{6}{1} times binom{5}{1} times 2 times frac{10!}{2! times 3!} )

c. 1 number repeated thrice with 7 or 8 included: ( binom{6}{1} times 2 times frac{10!}{4!} )

4. Including Number 7 or 8 Twice

We need to lay out the scenarios where the numbers 7 or 8 occur twice, but still ensure the condition is met.

a. 1 number repeated once with 7 or 8 included twice: ( binom{6}{1} times 2 times frac{10!}{2!^3} )

b. 1 number repeated twice with 7 or 8 included once: ( binom{6}{1} times frac{10!}{3!} )

c. 1 number repeated twice with 7 or 8 included twice: ( binom{6}{1} times 2 times frac{10!}{2!^3} )

d. 1 number repeated thrice with 7 or 8 included once: ( binom{6}{1} times 2 times frac{10!}{4!} )

5. Including Number 7 or 8 Thrice

The case where the number 7 or 8 is included thrice.

a. 2 numbers repeated once with 7 or 8 included thrice: ( binom{6}{2} times 2 times frac{10!}{2!^3} )

b. 1 number repeated once with 7 or 8 included thrice: ( binom{6}{1} times 2 times frac{10!}{3!} )

Add all these scenarios to get the total number of favorable outcomes. For simplicity, let's represent this mathematically:

Total favorable outcomes ( sum ) (all scenarios)

Total outcomes (8^{10})

Probability ( frac{text{Total favorable outcomes}}{8^{10}} )

Conclusion

The probability of rolling a number less than 7 on an 8-sided die is straightforward and can be derived as 0.75. However, if we extend this to rolled sequences of 10 times, the calculations become more complex and involve combinatorial analysis. This guide provides a detailed breakdown of the different scenarios and their corresponding probabilities.

The final probability for rolling 10 times and ensuring that the number is less than 7 in all outcomes can be approximated to 0.0987 as demonstrated by the calculations provided.