NCERT Class 11 Maths Chapter 16 Probability: Exercise 16.2

Understanding the Algebra of Events

In probability theory, a sample space $S$ represents all possible outcomes of a random experiment. Any subset $E$ of this sample space $S$ is defined as an event. Since events are simply subsets of a sample space, we can easily apply set-theoretic operations to combine and analyze them.

Here are the fundamental operations you need to know:

1. Complementary Event (Not A): Denoted by $A'$ or $A^c$, it includes all outcomes in the sample space $S$ that are not in event $A$. Mathematically, $A' = S - A$.
2. The Event 'A or B': Denoted by $A \cup B$ (Union). This represents outcomes that belong to either event $A$, event $B$, or both.
3. The Event 'A and B': Denoted by $A \cap B$ (Intersection). This represents outcomes common to both $A$ and $B$.
4. The Event 'A but not B': Denoted by $A - B$. This set contains outcomes that are in $A$ but strictly not in $B$, which is also written as $A \cap B'$.

Types of Events: Key Classifications

Simple (or Elementary) Event

An event that contains only a single sample point of the sample space. For example, when throwing a single die, the event of getting a '6' is {6}, which contains only one outcome.

Compound Event

An event that contains more than one sample point of the sample space. For example, getting an odd number on a die is {1, 3, 5}.

Mutually Exclusive Events

Two events A and B are mutually exclusive if the occurrence of one prevents the occurrence of the other. In other words, they have no outcomes in common: $A \cap B = \phi$.

Exhaustive Events

A group of events is exhaustive if their union covers the entire sample space. If $E_1 \cup E_2 \cup \dots \cup E_n = S$, then these events are called exhaustive.

Step-by-Step Solved Examples

• Example 1: Testing Mutually Exclusive and Exhaustive Events Problem: A single fair die is rolled. Let event $A$ be getting an odd number, and event $B$ be getting an even number. Determine if $A$ and $B$ are: (i) Mutually Exclusive, (ii) Exhaustive. Step 1: Identify the sample space: $S = \{1, 2, 3, 4, 5, 6\}$. Step 2: Write down events as subsets: $A = \{1, 3, 5\}$ and $B = \{2, 4, 6\}$. Step 3: Check for mutual exclusion by finding $A \cap B$. Here, $A \cap B = \{1, 3, 5\} \cap \{2, 4, 6\} = \phi$. Because the intersection is empty, $A$ and $B$ are mutually exclusive. Step 4: Check for exhaustiveness by finding $A \cup B$. Here, $A \cup B = \{1, 3, 5\} \cup \{2, 4, 6\} = \{1, 2, 3, 4, 5, 6\} = S$. Because their union equals $S$, the events are exhaustive.
• Example 2: Analyzing Three Coins Toss Problem: Three coins are tossed. Let $A$ be the event 'no heads show', $B$ be 'exactly one head shows', and $C$ be 'at least two heads show'. Show that $A$, $B$, and $C$ are mutually exclusive and exhaustive. Step 1: State the sample space: $S = \{HHH, HHT, HTH, THH, HTT, THT, TTH, TTT\}$. Step 2: List elements for each event: - $A = \{TTT\}$ - $B = \{HTT, THT, TTH\}$ - $C = \{HHT, HTH, THH, HHH\}$ Step 3: Test pair-wise intersections: - $A \cap B = \phi$ - $B \cap C = \phi$ - $A \cap C = \phi$ Since all pair-wise intersections are empty, the events are mutually exclusive. Step 4: Test union: $A \cup B \cup C = \{TTT, HTT, THT, TTH, HHT, HTH, THH, HHH\} = S$. Therefore, the events are also exhaustive.

Avoid This Common Board Exam Trap!

Many students confuse mutually exclusive events with independent events.

• Mutually Exclusive is a set-theoretic property: $A \cap B = \phi$ (they cannot happen at the same time).
• Independent Events is a probability-based property: $P(A \cap B) = P(A) \times P(B)$ (the occurrence of one doesn't affect the probability of the other).

Always double-check your definitions! In Exercise 16.2, your primary mathematical tool is set algebra. If you are asked to prove events are mutually exclusive, you must show that $A \cap B = \phi$ (or that their intersection contains 0 elements).

Practice Questions with Solutions

• Q: A die is rolled. Let $E$ be the event 'number is prime' and $F$ be the event 'number is even'. Are $E$ and $F$ mutually exclusive? A: Step 1: Write down the sample space for a die throw: $S = \{1, 2, 3, 4, 5, 6\}$. Step 2: List the outcomes for event $E$ (prime numbers): $E = \{2, 3, 5\}$. Step 3: List the outcomes for event $F$ (even numbers): $F = \{2, 4, 6\}$. Step 4: Find the intersection: $E \cap F = \{2, 3, 5\} \cap \{2, 4, 6\} = \{2\}$. Step 5: Check if the intersection is empty. Since $E \cap F \neq \phi$, the events are not mutually exclusive. Final answer: No, they are not mutually exclusive because they share the common outcome {2}.
• Q: Two coins are tossed. Let event $A$ be 'getting two heads' and event $B$ be 'getting two tails'. Check if these events are mutually exclusive and exhaustive. A: Step 1: Write the sample space: $S = \{HH, HT, TH, TT\}$. Step 2: Identify events: $A = \{HH\}$ and $B = \{TT\}$. Step 3: Check mutual exclusion: $A \cap B = \{HH\} \cap \{TT\} = \phi$. Thus, they are mutually exclusive. Step 4: Check exhaustiveness: $A \cup B = \{HH, TT\}$. Since $A \cup B \neq S$, they are not exhaustive. Final answer: The events are mutually exclusive but not exhaustive.
• Q: Write down the negation (complement) of the event 'getting at least one head' when rolling two coins. A: Step 1: Identify the sample space: $S = \{HH, HT, TH, TT\}$. Step 2: Write down the event $E$ ('getting at least one head'): $E = \{HH, HT, TH\}$. Step 3: Find the complement $E' = S - E$. Step 4: Subtraction yields: $E' = \{TT\}$. This corresponds to the event 'getting no heads' or 'getting two tails'. Final answer: $E' = \{TT\}$ (getting no heads).
• Q: Three events $P, Q, R$ are defined on a sample space such that $P = \{1, 2\}$, $Q = \{3, 4\}$, and $R = \{5, 6\}$ with $S = \{1, 2, 3, 4, 5, 6\}$. Categorize this system of events. A: Step 1: Check intersections: $P \cap Q = \phi$, $Q \cap R = \phi$, $P \cap R = \phi$. The events are pairwise mutually exclusive. Step 2: Check union: $P \cup Q \cup R = \{1, 2, 3, 4, 5, 6\} = S$. Step 3: Since both conditions are satisfied, we categorize them as mutually exclusive and exhaustive events. Final answer: They form a mutually exclusive and exhaustive set of events.

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