Chapter 13 ยท Introduction

Chapter 13 โ€” Probability

A probability is not fixed for ever: it changes the moment you learn something new. This chapter teaches you how to update probabilities with information, how to combine the stages of an experiment, how to reason backwards from an effect to its cause, and how to find the long-run average of a random quantity.

Before a one-day international begins, a commentator might say that the home side has about a $55\%$ chance of winning. Then the visitors win the toss and choose to bat on a flat pitch, and the number shifts. After $40$ overs they are $150$ for $7$, and the home side's chance is now perhaps $85\%$. Nothing random has been "re-rolled" โ€” the match is the same match โ€” but each new piece of information has cut down the set of ways the game can still go, and the probability has been recalculated inside that smaller set. That is conditional probability, the first and central idea of this chapter. From it come the multiplication theorem (how to find the chance that several things all happen, one stage at a time) and the precise meaning of independent events โ€” events for which the information about one tells you nothing about the other.

Often the question runs backwards. A blood test for a disease comes back positive: how likely is it that the patient really has the disease? An item from a factory is defective: which of the three machines most probably made it? Here we know the effect and want the probability of each possible cause. The theorem of total probability and Bayes' theorem answer such questions, and their answers often surprise people โ€” including doctors and judges. The chapter ends with random variables: numbers decided by chance, such as the number of sixes in three throws or a player's winnings in a game. Listing each value with its probability gives a probability distribution, and its mean $E(X)$ tells you what the quantity is worth "on average" over many repetitions โ€” the idea behind every fair game, insurance premium and expected-value decision.

Probability (Class 11)โ†’Conditional probabilityโ†’Multiplication theoremโ†’Independent eventsโ†’Total probabilityโ†’Bayes' theoremโ†’Random variables & $E(X)$
๐ŸŽฏ By the end of this chapter you canโ€ฆ
  • find a conditional probability $P(E\mid F)$ both from the formula and by shrinking the sample space to $F$, and use its properties;
  • use the multiplication theorem for two and three events, including drawing without replacement, and organise multi-stage experiments with a tree diagram;
  • decide whether two or three events are independent, and tell independence apart from mutual exclusiveness;
  • recognise a partition of a sample space and apply the theorem of total probability and Bayes' theorem to problems about causes and tests;
  • define a random variable, write down its probability distribution, and find its mean (expectation);
  • answer the Board's 1-, 2-, 3-, 5-mark and case-study questions on this chapter and the JEE Main / JEE Advanced problems built on it.
Teaser โ€” Stay or Switch?

A prize is hidden behind one of three doors. You pick a door. The host, who knows where the prize is, then opens one of the other doors that hides nothing, and offers you the chance to switch. Most people think it makes no difference โ€” two doors left, so $\tfrac12$ each. Press Play to run the game many times, with one player who always stays and one who always switches, and watch their winning rates settle. Can you explain the answer? Section 13.3 shows how.

games playedโ€”
stay: win rateโ€”
switch: win rateโ€”

Sections in this chapter

13.1

Conditional Probability

The probability of $E$ once we know that $F$ has happened: shrinking the sample space to $F$, the defining formula and its properties.

$P(E\mid F)=\dfrac{P(E\cap F)}{P(F)}$
Open section โ†’13.2

The Multiplication Theorem and Independent Events

Probabilities of "both" and "all three" stage by stage; tree diagrams; drawing with and without replacement; independent events.

$P(E\cap F)=P(E)\,P(F\mid E)$
Open section โ†’13.3

The Theorem of Total Probability and Bayes' Theorem

Partitions of a sample space; adding over all the causes; reasoning backwards from an observed effect to its most likely cause.

$P(E_i\mid A)=\dfrac{P(E_i)P(A\mid E_i)}{P(A)}$
Open section โ†’13.4

Random Variables and Probability Distributions

Numbers decided by chance, the table of their values and probabilities, and the mean (expectation) as a long-run average.

$E(X)=\sum x_ip_i$
Open section โ†’
โœ… Before you start
This chapter builds directly on Class 11 Mathematics, Ch. 14 โ€” Probability: random experiments, sample spaces and events, the algebra of events ($A\cup B$, $A\cap B$, $A'$), mutually exclusive and exhaustive events, the axioms of probability, $P(E)=\dfrac{n(E)}{n(S)}$ for equally likely outcomes, $P(A')=1-P(A)$ and the addition rule $P(A\cup B)=P(A)+P(B)-P(A\cap B)$. For problems about cards, committees and balls drawn from bags you also need combinations $^nC_r$ from Class 11 Mathematics, Ch. 7 โ€” Permutations and Combinations. The idea of splitting a set into non-overlapping pieces that cover it โ€” a partition โ€” was met with equivalence classes in Section 1.1. Quick self-test: two fair dice are thrown; what is the probability that the sum is $7$? In how many ways can $2$ cards be chosen from a pack of $52$? (Answers: $\tfrac{6}{36}=\tfrac16$ and $^{52}C_2=1326$.)
โ† 12.3 Unbounded and Empty Feasible Regions13.1 Conditional Probability โ†’