As your math teacher, let's solve this problem together step by step.
a. Theoretical Probability of Rolling a 3:
A standard number cube, often referred to as a die, has six faces numbered 1 through 6. Since all faces are usually designed to be equally probable, the theoretical probability of rolling a three (or any specific number) is the ratio of the number of successful outcomes (rolling a 3) to the total number of possible outcomes (any number from 1 to 6).
So,
Theoretical Probability = Number of successful outcomes / Total number of possible outcomes
= 1 / 6
This is already in the simplest form since 1 and 6 have no common divisors other than 1.
b. Experimental Probability of Rolling a 3:
The experimental probability, in contrast to the theoretical probability, is determined based on actual trials or experiments. In this case, the cube was rolled 450 times, and the number 3 appeared 67 times.
So,
Experimental Probability = Number of times a 3 was rolled / Total number of rolls
= 67 / 450
Now, to simplify the fraction, you would find the greatest common divisor (GCD) of 67 and 450 and divide both numerator and denominator by that GCD.
Looking at the numbers, they do not immediately suggest a common factor, and since 67 is a prime number, we can reasonably conclude that the fraction is already in its simplest form. Therefore, the experimental probability is:
= 67 / 450
So there you have it, answers to both questions in their simplest form:
a. The theoretical probability of rolling a 3 on a number cube is 1/6.
b. The experimental probability of rolling a 3 given the cube is rolled 450 times and 3 comes up 67 times is 67/450.