Final answer:
The odds against rolling a number less than 7 on a fair eight-sided die are 1 to 3, as there are 6 favorable outcomes (numbers 1-6) and two unfavorable outcomes (numbers 7 and 8).
Step-by-step explanation:
The question involves calculating the odds against rolling a number less than 7 on a fair eight-sided die. In this scenario, we consider the numbers 1 through 6 as favorable outcomes since they are less than 7. Therefore, the probability of rolling a number less than 7, P(E), is the number of favorable outcomes divided by the total number of possible outcomes which are 8. So, P(E) = 6/8 which simplifies to 3/4 as our favorable probability.
To determine the odds against E, we need to compare the probability of the event not occurring to the probability of the event occurring. The probability of not rolling a number less than 7 is the probability of rolling either a 7 or an 8, which is 2/8 or 1/4. The odds against an event are the ratio of the probability of the event not occurring to the probability of the event occurring. Therefore, the odds against rolling a number less than 7 are 1/4 divided by 3/4, which simplifies to 1/3, or in other terms, 1 to 3.