Lesson 1 of 11 · 7 min

Events and sample spaces

NCERT §14.1

The Class XI coin stall at the mela has one rule: toss three coins, and if at least one head shows, you win a sweet. Before anyone plays, a teacher asks the stall team to write down every way the three coins can land. How many are there, and which of them win?

The story this chapter follows: The Class XI mela stalls

Imagine the school mela, where Class XI runs a row of game stalls: a three-coin toss, a trick die, a token bag, a card stall, a two-dice stall and a lucky draw. Every stall, rule and number in this chapter is made up for illustration, chosen so the counting comes out clean.

The lesson in notes

In short

A random experiment has more than one possible outcome, and which one will turn up cannot be predicted in advance. The set of all its outcomes is the sample space S; it plays the role of the universal set for every question about that experiment.

Toss a coin twice: S = {HH, HT, TH, TT}. The outcomes showing exactly one head are HT and TH, so that happening corresponds to the subset E = {HT, TH} of S.

Every description of what might happen picks out a subset of S. For the same two tosses: 'exactly two tails' is {TT}; 'at least one tail' is {HT, TH, TT}; 'at most one head' is also {HT, TH, TT}; 'second toss is not a head' is {HT, TT}.

Two differently worded descriptions can give the same subset, as 'at least one tail' and 'at most one head' do above. They are then the same event.

'Number of tails is at most two' takes in all of S, while 'more than two tails' takes in nothing, the empty set φ.

Definition: an event is any subset E of the sample space S. A sample space with n outcomes therefore has 2ⁿ events, counting φ and S.

Made-up illustration used through these notes: at a school mela, one stall tosses three coins. Its sample space has 2 × 2 × 2 = 8 outcomes: HHH, HHT, HTH, THH, HTT, THT, TTH, TTT.