Mathematics, Statistics & Data Science
High School (11–12)
Statistics and Probability › ProbabilityChapter 23Introduction to Probability
Before a match, a referee tosses a coin.
- Before a board game, you throw a die.
- You cannot say what will happen, but you can list what might happen, and then the question about chance becomes a question about counting.
- This chapter starts with the list of all outcomes and the events built from it.
- It then finds probabilities by counting, with the addition theorem, cards, committees and arrangements.
- It ends with the problems that exams love: sums of dice, birthdays, matching letters to envelopes and a point chosen at random on a line.
- Python checks every probability by listing the outcomes.
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- It covers the unit of the Indian syllabus (NCERT, JEE Main) and the matching topics of A-Level, IB and AP.
Part of Mathematics · High School (11–12) (chapter 23)
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Sample Spaces and Events
A bag holds a red, a green and a blue ball, and you take out two. How many results are possible? The answer depends on whether you put the first ball back, and on whether the order matters. Learn to write the list of all outcomes for coins, dice, cards and bags, to name the kinds of event, and to combine events with union, intersection and complement. It covers the first part of the Indian syllabus unit (NCERT, JEE Main) and the matching topics of A-Level, IB and AP.
Probability by Counting
What is the chance that a hand of five cards holds exactly two aces? You could list every hand, but there are more than two and a half million of them. Count them with combinations instead. Learn the classical definition and its rules, the addition theorem for two and three events, odds and the trick of the complement for at least one. Then use the counting tools of the permutations and combinations chapter on cards, committees and arrangements. It covers the second part of the Indian syllabus unit (NCERT, JEE Main) and the matching topics of A-Level, IB and AP.