The Mathematics of Maybe: Introduction to Probability | Exercise 7.2

Question 6

Suppose you roll a 6-sided die 12 times and get a '3' three times.

(i) What is the experimental probability of rolling a '3'?

(ii) What is the theoretical probability of rolling a '3'?

(iii) Why might these probabilities be different? What would you expect to happen if you roll the die 60, 600, or 6000 times?

Solution

Understand the Question

Experimental Probability: Calculated from actual observed results of an experiment: $P(E) = \dfrac{\text{Number of trials where the event occurred}}{\text{Total number of trials}}$
Theoretical Probability: Calculated based on all possible equally likely outcomes without conducting an experiment: $P(E) = \dfrac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$
For a small number of trials, chance variation causes experimental results to differ from theoretical expectations. As the number of trials increases, the experimental probability gets closer to the theoretical probability.

(i) What is the experimental probability of rolling a '3'?

Step 1 · Calculate Experimental Probability

Diagram 1

Given: $\text{Total number of rolls} = 12$ $\text{Number of times '3' appeared} = 3$

\begin{aligned} P(\text{rolling a '3' experimentally}) &= \dfrac{\text{Number of times '3' appeared}}{\text{Total number of rolls}} \\[0.6em] &= \dfrac{3}{12} \\[0.6em] &= \dfrac{1}{4} \end{aligned}

Answer

(i) $\dfrac{1}{4}$

(ii) What is the theoretical probability of rolling a '3'?

Step 1 · Calculate Theoretical Probability

A standard 6-sided die has $6$ possible outcomes: $\{1, 2, 3, 4, 5, 6\}$ . Only $1$ face corresponds to rolling a '3'.

\begin{aligned} P(\text{rolling a '3' theoretically}) &= \dfrac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \\[0.6em] &= \dfrac{1}{6} \end{aligned}

Answer

(ii) $\dfrac{1}{6}$

(iii) Why might these probabilities be different? What would you expect to happen if you roll the die 60, 600, or 6000 times?

Step 1 · Explain Difference and Effect of Increasing Trials

Reason for difference: Experimental probability is based on actual trials from a small sample ( $12$ rolls), which naturally fluctuates due to random chance. Theoretical probability assumes an ideal, perfectly fair die where each face is equally likely.
Increasing trials: According to the Law of Large Numbers, as the number of trials increases ( $60$ , $600$ , or $6000$ rolls), the experimental probability gets closer and closer to the theoretical probability of $\dfrac{1}{6}$ .

Answer

(iii) The probabilities differ due to random chance in a small number of trials. As the number of rolls increases to 60, 600, or 6000, the experimental probability gets closer and closer to $\dfrac{1}{6}$ .

Common Mistakes

Confusing Theoretical and Experimental Probability: Calculating theoretical probability using the experiment's results instead of the sample space of a standard die ( $\dfrac{1}{6}$ vs $\dfrac{3}{12}$ ).
Small Sample Assumption: Assuming experimental probability must always match theoretical probability even for very small numbers of trials.

Question 6

More questions in Exercise 7.2