Using data; the long run

Probability from data. If 20 of 50 students like mango best, the chance that a randomly chosen student does is 20/50 = 0.4.

Sample and population. We usually collect data from a sample (a smaller group), not the whole population. With 0.4 from a class of 50, a school of 1500 needs about 0.4 × 1500 = 600 mangoes. A bigger sample, drawn from different classes, makes the estimate more trustworthy. This is sampling.

Experiment and theory can disagree, especially with few trials. Three 3s in 12 rolls gives 1/4, not 1/6. That is not because the die is unfair: 12 rolls are simply too few.

The Law of Large Numbers: as the number of trials grows, the experimental probability gets closer and closer to the theoretical one. With 600 or 6000 rolls you would expect close to 1/6.

The Gambler's Fallacy: a coin has no memory. After six heads in a row, tails is not 'due': the next toss is still 1/2. After three 6s, the next roll is still 1/6.

Probability does not tell you what will happen next; it tells you what happens in the long run.