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

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

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

(C) Why might these probabilities be different? What would you expect to happen if you roll the die 60, 600, or 6000 times?                                                                                                                                                                                                [Page No. 166]

(A) Experimental probability, $P(3) = \frac{3}{12} = \frac{1}{4}$

(B) Theoretical probability, $P(3) = \frac{1}{6}$

(C) The two probabilities differ because 12 trials is a small number. Random variation (luck) causes the observed frequency to deviate from the theoretical value.

As we increase the number of trials:

• In 60 rolls, we would expect around $60 \times \frac{1}{6} = 10$ threes, and observed values should cluster near 10.

• In 600 rolls, we would expect around 100 threes, and the experimental probability would be very close to $\frac{1}{6}$ .

• In 6000 rolls, the experimental probability would be essentially close to $\frac{1}{6}$ .

This is the Law of Large Numbers: as trials increase, experimental probability approaches theoretical probability.

Caution
Students should remember that experimental and theoretical probability are not always equal. They differ for a small number of trials and converge only as the number of trials becomes large