Question 6

We will find the experimental and theoretical probabilities of rolling a '3'.

Step 1 — Calculate experimental probability

Experimental probability uses actual results. We rolled the die 12 times. We got a '3' three times.

$P(\text{rolling a '3' experimentally}) = \frac{\text{Number of times '3' appeared}}{\text{Total number of rolls}}$

$= \frac{3}{12}$

$= \frac{1}{4}$

$\boxed{P(\text{3}) \text{ experimentally} = \frac{1}{4}}$

Step 2 — Calculate theoretical probability

Theoretical probability is based on possibilities. A standard die has 6 faces. Each face has an equal chance of appearing. Only one face shows a '3'.

$P(\text{rolling a '3' theoretically}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$

$= \frac{1}{6}$

$\boxed{P(\text{3}) \text{ theoretically} = \frac{1}{6}}$

Step 3 — Explain differences and future expectations

The probabilities are different. Experimental probability comes from a small number of trials. It can vary a lot due to chance. Theoretical probability is what we expect from a fair die. It assumes all outcomes are equally likely.

As we roll the die more times, the experimental probability gets closer to the theoretical probability. This is called the Law of Large Numbers. If we roll 60 times, the experimental probability will be closer to 1/6. If we roll 600 times, it will be very close to 1/6. If we roll 6000 times, it will be almost exactly 1/6.

Answer

(i) The experimental probability of rolling a '3' is $\frac{1}{4}$. (ii) The theoretical probability of rolling a '3' is $\frac{1}{6}$. (iii) The probabilities are different because experimental probability uses actual results from a limited number of trials, which can vary by chance. Theoretical probability is based on an ideal, fair die. As the number of rolls increases (e.g., 60, 600, 6000 times), the experimental probability would get closer and closer to the theoretical probability of $\frac{1}{6}$.