Classical probability: sample space and events

What it shows

The classical definition of probability applies when all outcomes of a trial are equally likely: P(A) = n(A)/n(Ω), where Ω is the sample space. The crucial step is listing Ω correctly. Tossing two coins gives four ordered outcomes, HH, HT, TH and TT, so one head and one tail has probability 2/4, not 1/3. Rolling two dice gives 36 ordered pairs, laid out as a 6 × 6 grid. The simulation highlights the outcomes in A, counts them, and can repeat the trial N times to compare the frequency with the probability.

How to use

Choose a Trial and an Event A from the list, setting the Value of k where it is needed. Before reading P(A), ask students to predict n(A) and count the highlighted cells. Use Custom (click cells) to build events of their own, such as the complement of a given event. Then set the Number of trials N and press Run N trials to compare the frequency with the theoretical probability.

Parameters you can change

• Trial Toss 2 coins, Toss 3 coins, Roll 1 die, Roll 2 dice
• Event A At least one head / one six (1 die: the number is prime), Exactly k heads / sum of pips equals k / number equals k, Number of heads is even / sum of pips is even / number is even, All faces match (1 die: the number is divisible by 3), Custom, chosen by clicking cells
• Value of k 0–12
• Number of trials N 100–10000 trials

Questions to explore

1. When two coins are tossed, why is the probability of one head and one tail 1/2 rather than 1/3?
2. Which sum of two dice is most likely, and how many outcomes in the grid give it?
3. For three coins, what is the complement of “at least one head”, and what is its probability?