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Probability - Types of events

Grade 11CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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An event EE is a subset of the sample space SS. Every outcome of a random experiment is an element of SS, and an event is a collection of some of these outcomes.

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Impossible Event: An event that contains no outcomes from the sample space, denoted by ∅\emptyset. Its probability is P(∅)=0P(\emptyset) = 0.

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Sure (Certain) Event: An event that contains all possible outcomes of the sample space SS. Its probability is P(S)=1P(S) = 1.

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Simple (Elementary) Event: An event that has only one sample point (outcome) of the sample space. For example, in tossing a coin, {H}\{H\} is a simple event.

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Compound Event: An event that has more than one sample point. For example, in rolling a die, the event 'getting an even number' E={2,4,6}E = \{2, 4, 6\} is a compound event.

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Complementary Event: For any event AA, the event 'not AA' is called the complementary event, denoted by A′A' or Aˉ\bar{A}. It contains all outcomes in SS that are not in AA.

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Mutually Exclusive Events: Two events AA and BB are mutually exclusive if the occurrence of one excludes the occurrence of the other. Mathematically, A∩B=∅A \cap B = \emptyset.

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Exhaustive Events: Events E1,E2,...,EnE_1, E_2, ..., E_n are exhaustive if their union is the entire sample space, i.e., E1∪E2∪...∪En=SE_1 \cup E_2 \cup ... \cup E_n = S.

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Equally Likely Events: Events are said to be equally likely if none of them is expected to occur in preference to the others (e.g., getting a Head or a Tail on a fair coin).

📐Formulae

P(E)=n(E)n(S)P(E) = \frac{n(E)}{n(S)}

P(A′)=1−P(A)P(A') = 1 - P(A)

For mutually exclusive events A and B:P(A∩B)=0\text{For mutually exclusive events } A \text{ and } B: P(A \cap B) = 0

For exhaustive events A and B:P(A∪B)=1\text{For exhaustive events } A \text{ and } B: P(A \cup B) = 1

P(A∪B)=P(A)+P(B)−P(A∩B)P(A \cup B) = P(A) + P(B) - P(A \cap B)

💡Examples

Problem 1:

A fair die is rolled. Let event AA be 'getting an even number' and event BB be 'getting an odd number'. Determine if AA and BB are mutually exclusive and exhaustive.

Solution:

The sample space is S={1,2,3,4,5,6}S = \{1, 2, 3, 4, 5, 6\}. \nEvent A={2,4,6}A = \{2, 4, 6\}. \nEvent B={1,3,5}B = \{1, 3, 5\}. A∩B=∅A \cap B = \emptyset (since no element is common), therefore they are mutually exclusive. A∪B={1,2,3,4,5,6}=SA \cup B = \{1, 2, 3, 4, 5, 6\} = S, therefore they are exhaustive.

Explanation:

Since the intersection is empty and the union equals the sample space, the events satisfy both definitions.

Problem 2:

Two coins are tossed simultaneously. Find the probability of the event 'getting at least one head'.

Solution:

The sample space is S={HH,HT,TH,TT}S = \{HH, HT, TH, TT\}, so n(S)=4n(S) = 4. \nLet EE be the event of getting at least one head. E={HH,HT,TH}E = \{HH, HT, TH\}, so n(E)=3n(E) = 3. P(E)=n(E)n(S)=34P(E) = \frac{n(E)}{n(S)} = \frac{3}{4}

Explanation:

The event 'at least one head' includes outcomes with exactly one head and outcomes with two heads.

Problem 3:

In a single throw of a die, find the probability of getting a number less than 77.

Solution:

The sample space S={1,2,3,4,5,6}S = \{1, 2, 3, 4, 5, 6\}, so n(S)=6n(S) = 6. \nLet EE be the event of getting a number less than 77. E={1,2,3,4,5,6}=SE = \{1, 2, 3, 4, 5, 6\} = S. P(E)=66=1P(E) = \frac{6}{6} = 1

Explanation:

This is a 'Sure Event' because every possible outcome in the sample space satisfies the condition.