Question

A fair die with its faces numbered from 1 to 6 will be rolled. Which of the following is the best interpretation of the probability that the number landing face up will be less than 3?

A. For many rolls of the die, the long-run relative frequency of a number less than 3 landing face up is 1/3.
B. For many rolls of the die, the long-run relative frequency of a number less than 3 landing face up is 1/2.
C. For many rolls of the die, the long-run relative frequency of a number less than 3 landing face up is 2/3.
D. For three rolls of the die, a number less than 3 will land face up one time.
E. It will take three rolls of the die before a number less than 3 lands face up for the first time.

Answer 1

The best interpretation of the probability of rolling a number less than 3 on a fair six-sided die is that in the long-run relative frequency, such a number is expected to land face up approximately 1/3 of the time, aligning with the law of large numbers. The correct option is A.

Step-by-step explanation:

The probability that the number landing face up on a fair six-sided die is less than 3 involves considering the outcomes 1 and 2, as these are the only numbers less than 3 on the die. There are 6 equally possible outcomes when rolling a fair die, so the probability of rolling a number less than 3 is the number of favorable outcomes (2) divided by the total possible outcomes (6), which simplifies to 1/3.

Therefore, the best interpretation of the probability that the number landing face up will be less than 3 is: A. For many rolls of the die, the long-run relative frequency of a number less than 3 landing face up is 1/3. This means that if you were to roll the die a large number of times, you would expect, on average, for a number less than 3 to come up about one-third of the time.

This is an application of the law of large numbers, which states that as the number of trials increases, the experimental probability tends to approach the theoretical probability.