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 \\( \frac { 1 } { 3 } \\).
b for many rolls of the die, the long - run relative frequency of a number less than 3 landing face up is \\( \frac { 1 } { 2 } \\).
c for many rolls of the die, the long - run relative frequency of a number less than 3 landing face up is \\( \frac { 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.

Explanation:

Step1: Determine the total number of outcomes

A fair die has 6 faces, so the total number of possible outcomes when rolling the die is \(n = 6\).

Step2: Determine the number of favorable outcomes

The numbers less than 3 on a die are 1 and 2. So the number of favorable outcomes \(m=2\).

Step3: Calculate the probability

The probability \(P\) of an event is given by the formula \(P=\frac{m}{n}\). Substituting \(m = 2\) and \(n = 6\), we get \(P=\frac{2}{6}=\frac{1}{3}\). Probability represents the long - run relative frequency.

Answer:

A. For many rolls of the die, the long - run relative frequency of a number less than 3 landing face up is \(\frac{1}{3}\)