What went wrong when the king rode under the arch?

It was too low and knocked off his crown

How does the chain of blame unfold in the poem?

Each accused person blames another, ending with the king's own changes

What is an event in probability?

Any subset \(E\) of the sample space \(S\) of a random experiment.

Define the impossible and sure events.

The empty set \(\varphi\) is the impossible event; the whole sample space \(S\) is the sure event.

When are two events mutually exclusive?

When they cannot occur simultaneously, i.e. \(A\cap B=\varphi\) (disjoint sets).

State the axioms of probability.

For every event \(E\), \(P(E)\ge0\); \(P(S)=1\); and for mutually exclusive \(E,F\), \(P(E\cup F)=P(E)+P(F)\). It follows that \(P(\varphi)=0\).

Give the probability for equally likely outcomes.

\(P(A)=\dfrac{n(A)}{n(S)}=\dfrac{\text{favourable outcomes}}{\text{total outcomes}}\).

State the addition rule for any two events.

\(P(A\cup B)=P(A)+P(B)-P(A\cap B)\); if mutually exclusive, \(P(A\cup B)=P(A)+P(B)\).

State the complement rule.

\(P(\text{not }A)=P(A')=1-P(A)\).

One card is drawn from 52. Find the probability it is a diamond.

There are 13 diamonds, so \(P=\dfrac{13}{52}=\dfrac14\).

A coin is tossed twice. Find the probability of at least one tail.

Sample space \(\{HH,HT,TH,TT\}\); favourable \(=3\), so \(P=\dfrac34\).

When are events exhaustive and mutually exclusive together?

When \(E_1\cup E_2\cup\cdots\cup E_n=S\) and \(E_i\cap E_j=\varphi\) for all \(i\ne j\).

Write the formula for the theoretical probability of an event E.

\(P(E) = \dfrac{\text{number of outcomes favourable to } E}{\text{total number of equally likely outcomes}}\).

What is the probability of a sure event and of an impossible event?

\(1\) for a sure (certain) event and \(0\) for an impossible event.

What is the range of the probability of any event?

The probability of any event lies between 0 and 1, that is \(0 \le P(E) \le 1\).

What are complementary events, and how are their probabilities related?

An event \(E\) and "not \(E\)" are complementary; \(P(E) + P(\text{not }E) = 1\), so \(P(\text{not }E) = 1 - P(E)\).

A die is thrown once. What is the probability of getting an even number?

Favourable outcomes are 2, 4, 6 (three of them) out of six, so \(P = \dfrac{3}{6} = \dfrac{1}{2}\).

A card is drawn from a well-shuffled pack of 52. What is the probability that it is a king?

There are 4 kings, so \(P(\text{king}) = \dfrac{4}{52} = \dfrac{1}{13}\).

Two coins are tossed together. What is the probability of getting two heads?

The outcomes are HH, HT, TH, TT; only one is favourable, so \(P(\text{two heads}) = \dfrac{1}{4}\).

If the probability of an event is 0.7, what is the probability of its not happening?

\(P(\text{not }E) = 1 - 0.7 = 0.3\).

Probability

Class 11 Mathematics · Chapter 14

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Summary

Building on random experiments and sample spaces, this chapter takes the axiomatic approach to probability. Any subset \(E\) of the sample space \(S\) is an event; \(\varphi\) is the impossible event and \(S\) the sure event. Events combine like sets: \(A\cup B\) is "A or B", \(A\cap B\) is "A and B", \(A'\) is "not A", and \(A-B\) is "A but not B". Two events are mutually exclusive if \(A\cap B=\varphi\), and a collection is exhaustive if its union is \(S\). Probability \(P\) assigns to each event a number satisfying \(P(E)\ge0\), \(P(S)=1\), and \(P(A\cup B)=P(A)+P(B)\) for mutually exclusive events, from which \(P(\varphi)=0\). For equally likely outcomes \(P(A)=\dfrac{n(A)}{n(S)}\). General results include the addition rule \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\) and the complement rule \(P(\text{not }A)=1-P(A)\). These let us compute chances for coins, dice, cards and selections.

Events and types of eventsAlgebra of eventsMutually exclusive and exhaustive eventsAxiomatic approach to probabilityProbability of equally likely outcomesAddition and complement rules

Key terms

Event

Any subset \(E\) of the sample space \(S\).

Impossible and sure events

\(\varphi\) is impossible; \(S\) is the sure event.

Mutually exclusive events

Events with \(A\cap B=\varphi\) that cannot occur together.

Exhaustive events

Events whose union is the whole sample space \(S\).

Equally likely outcomes

Outcomes with the same probability, so \(P(A)=\dfrac{n(A)}{n(S)}\).

Addition rule

\(P(A\cup B)=P(A)+P(B)-P(A\cap B)\).

Extra questions & answers

Try one from this chapter

An event in probability is:

Flashcards & interactive practice