Suppose you roll a 6-sided die 12 times and get a '3' three times. Class 9
Suppose you roll a 6-sided die 12 times and get a '3' three times. Class 9
Question 1.
Suppose you roll a 6-sided die 12 times and get a '3' three times. Class 9
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:
i. Number of times 3 occurred = 3
Total number of trials = 12
$\begin{aligned} & \mathrm{P}(\mathrm{E})=\frac{\text { Number of times event occurred }}{\text { Total number of trials }} \\ & \mathrm{P}(\text { rolling a } 3)=\frac{3}{12}=\frac{1}{4}=0.25\end{aligned}$
∴ The experimental probability of rolling a 3 is 0.25.
ii. All possible outcomes = 1, 2, 3, 4, 5, 6
Number of all possible outcomes = 6
Number of favourable outcomes (getting 3) = 1
$\begin{aligned} \mathrm{P}(\text { rolling a } 3) & =\frac{\text { Number of favourable outcomes }}{\text { Total number of outcomes }} \\ & =\frac{1}{6}=0.167\end{aligned}$
∴ The theoretical probability of rolling a 3 is 0.167.
iii. The probabilities are different because experimental probability is based on observed outcomes in only 12 trials, which is a small sample, whereas the theoretical probability assumes perfect fairness (all outcomes are equally likely).
As the number of rolls increases (60, 600, 6000), the experimental probability is approaches the theoretical probability according to the law of large numbers.
When we used the sample space {Rain, No Rain} in Example 1, we focused only on whether it will rain or not. However, if we want to include different amounts of rainfall like drizzle, light rain or heavy rain, we need to expand the sample space to {No Rain, Drizzle, Light Rain, Heavy Rain} so that it better matches the level of detail required for the question. It is important to ensure the sample space is detailed enough to suit the specific problem being studied.