ML Aggarwal Class 11 Maths Solutions Chapter 16 Probability

PROBABILITY

Introduction

The opening statements - "It may rain today", "Rajesh is quite sure to top his class", "It is highly unlikely that Salman will marry Kareena", "I have bet 100 rupees on India winning against Pakistan" - all show situations with uncertain outcomes in daily life. Probability serves as a way to measure this uncertainty. The field of probability emerged from examining games of chance and gambling. Think about this scenario: you pay Rs 2 to draw a single card from a standard 52-card deck. If you pull an ace, you receive Rs 20; otherwise, nothing. Would you participate in this game? Does it offer fair odds, or will you end up losing money over time? Questions like these sparked the development of modern probability theory. Key contributors to this field include Jerome Cardan (Italian), Pascal and Pierre de Fermat (French), James Bernoulli (Swiss), and Kolmogorov (Russian).

16.1 Random Experiments and Sample Spaces

Experiment

An action or operation that yields some definite outcomes is known as an experiment. At times, the result is certain and predictable. For instance, if a triangle is provided without knowing its angles, we can always confirm that the sum of its three angles equals 180°. These are called deterministic experiments. Other times, the outcome is not fixed. For example, flipping a coin may land as heads or tails. Such experiments are called probabilistic or random.

Random Experiment

An experiment is termed random if it can result in two or more distinct outcomes, and we cannot predict which outcome will occur beforehand. For an experiment to be classified as random, it must satisfy two criteria:

• It must have more than one possible outcome.
• It must be impossible to predict the outcome in advance.

Examples of random experiments include tossing a coin, tossing two coins at the same time, tossing a coin three times, throwing a die, and drawing a card from a 52-card deck.

Sample Space

The set of all possible outcomes that can occur from a random experiment is called the sample space, generally denoted by S. Each individual outcome in this set is called a sample point. When a coin is tossed once, the possible outcomes are heads or tails. Using H to denote heads and T to denote tails, the sample space is S = {H, T}. When a die is rolled, it can land showing any of the six numbers 1, 2, 3, 4, 5, or 6. Therefore, the sample space is S = {1, 2, 3, 4, 5, 6}.

Example 1. Two coins (a one-rupee coin and a two-rupee coin) are tossed together. Describe the sample space.
Answer: Each coin can come up heads (H) or tails (T). Since the coins are distinguishable, we represent outcomes using ordered pairs. The four possible outcomes are (H, H) if both show heads, (H, T) if the first shows head and second shows tail, (T, H) if the first shows tail and second shows head, and (T, T) if both show tails. Thus the sample space is S = {(H, H), (H, T), (T, H), (T, T)}, which can also be written as S = {HH, HT, TH, TT}.
In simple words: When two coins are tossed, there are four ways they can land - both heads, first head then tail, first tail then head, or both tails.

Exam Tip: Use ordered pairs for distinguishable objects (like a one-rupee and two-rupee coin) to show which outcome belongs to which coin.

Example 2. Describe the sample space of a random experiment when a coin and a die are tossed together.
Answer: The sample space has 2 × 6 = 12 possible outcomes. S = {(H, 1), (H, 2), (H, 3), (H, 4), (H, 5), (H, 6), (T, 1), (T, 2), (T, 3), (T, 4), (T, 5), (T, 6)}. This can alternatively be written as S = {H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}.
In simple words: The coin can land two ways (heads or tails) and the die can show six numbers. Multiply these to get 12 total combinations.

Exam Tip: For independent experiments (coin and die), count possibilities separately and multiply them together to find the total number of outcomes.

Example 3. Find the sample space associated with the experiment of rolling a pair of dice (one is blue and the other red) once. Also find the number of elements of the sample space.
Answer: The blue die may show any number from 1 to 6, and the red die may also show any number from 1 to 6. We denote each outcome as an ordered pair (x, y) where x is the number on the blue die and y is the number on the red die. The sample space is S = {(x, y); x is a number on the blue die, and y is a number on the red die}. Listing all possibilities:

The total number of outcomes is 6 × 6 = 36.
In simple words: With two dice each showing one of six numbers, you get 36 different possible pairs when you multiply 6 times 6.

Exam Tip: For dice problems, use a table to systematically list all outcomes - this ensures no pair is missed and helps verify your count.

Example 4. An experiment consists of recording boy-girl composition of families with 2 children. (i) What is the sample space if we are interested in knowing whether the elder child is a boy or a girl? (ii) What is the sample space if we are interested in knowing whether it is a boy or a girl in the order of their births? (iii) What is the sample space if we are interested in number of girls in the family?
Answer:

(i) Since only the elder child's gender matters, there are just two possible outcomes. The sample space is S = {Boy, Girl} or S = {B, G}.

(ii) With two children, considering birth order, four outcomes are possible: both boys, boy then girl, girl then boy, both girls. The sample space is S = {BB, BG, GB, GG}.

(iii) If we only care about counting girls, there are three possible outcomes: zero girls, one girl, or two girls. The sample space is S = {0, 1, 2}.
In simple words: The sample space depends on what information you track - just the elder child, birth order, or only the count of girls.

Exam Tip: Always identify what the question is asking about - this determines which outcomes to include in your sample space.

Example 5. An experiment consists of tossing a coin and throwing it second times if a head occurs. If a tail occurs on the first toss, then a die is rolled. Find the sample space.
Answer: A coin is tossed first. If heads appears, the coin is tossed again. If tails appears on the first toss, a die is rolled instead. Using a tree diagram to map out all paths:

Reading from the tree, the possible outcomes are HH (heads on first, heads on second), HT (heads on first, tails on second), T1, T2, T3, T4, T5, T6 (tail on first, then die shows that number). Therefore, the sample space is {HH, HT, T1, T2, T3, T4, T5, T6}, containing 8 possible outcomes.
In simple words: After the first coin toss, the next step depends on the result - if heads, toss again; if tails, roll a die instead.

Exam Tip: Use tree diagrams for conditional experiments (where later actions depend on earlier results) to systematically trace all possible paths and outcomes.

Example 6. A coin is tossed. If it shows a tail, we draw a ball from a box which contains 2 red and 3 black balls. If it shows head, we throw a die. Find the sample space for this experiment.
Answer: Label the red balls as R1 and R2, and the black balls as B1, B2, and B3. Based on the coin outcome, either a die is thrown or a ball is drawn:

If the coin shows heads, the die is thrown, giving outcomes H1, H2, H3, H4, H5, H6. If the coin shows tails, a ball is drawn, giving outcomes TR1, TR2, TB1, TB2, TB3. Therefore, the sample space is {H1, H2, H3, H4, H5, H6, TR1, TR2, TB1, TB2, TB3}, containing 11 possible outcomes.
In simple words: The coin result determines what happens next - either roll a die or pick a ball.

Exam Tip: Label objects clearly (like R1, R2 for red balls and B1, B2, B3 for black balls) and organize outcomes by the initial branch (H or T) to ensure complete and accurate listing.

Example 7. A coin is tossed twice. If the second throw results in a tail, we roll a die. Describe the sample space. How many outcomes are possible in this experiment?
Answer: Tracing through the decision tree: if the first toss is heads, then on the second toss - if heads appears, the experiment ends with outcome HH; if tails appears, roll a die to get HT1 through HT6. If the first toss is tails, on the second toss - if heads appears, the experiment ends with outcome TH; if tails appears, roll a die to get TT1 through TT6. Combining all paths, the sample space is S = {HH, HT1, HT2, HT3, HT4, HT5, HT6, TH, TT1, TT2, TT3, TT4, TT5, TT6}. There are 14 possible outcomes.
In simple words: The first coin toss gives two paths. In each path, the second toss decides - if tails, roll the die; if heads, stop.

Exam Tip: Carefully track conditional branches in multi-step experiments by following each decision path from start to finish before combining all outcomes.

Example 8. A coin is tossed. If the result is a head, a die is thrown. If the die shows up the numbers 1 or 3, the die is thrown again and if it shows the number 5, then a coin is tossed. Write the sample space of this experiment. How many outcomes are possible in this experiment?
Answer: When the coin is tossed, if tails shows, the experiment finishes with outcome T. If heads shows, a die is thrown. If the die displays an even number (2, 4, 6), the experiment ends with outcomes H2, H4, or H6. If the die shows 1 or 3, the die is thrown a second time, producing outcomes H1 followed by a second die result (H11, H12, H13, H14, H15, H16) or H3 followed by a second die result (H31, H32, H33, H34, H35, H36). If the die shows 5, a coin is tossed, giving outcomes H5H or H5T. Therefore, the sample space is S = {T, H2, H4, H6, H11, H12, H13, H14, H15, H16, H31, H32, H33, H34, H35, H36, H5H, H5T}, with 18 possible outcomes.
In simple words: The initial coin and die results branch into different paths. Some branches end immediately (even die numbers), while others continue with more steps (odd numbers or 5).

Exam Tip: For complex nested experiments, build the sample space branch by branch, making sure each path is followed completely before moving to the next one to avoid duplicates or omissions.

Example 9. Consider the experiment in which a coin is tossed repeatedly until a head comes up for the first time. Describe the sample space. What is the number of possible outcomes?
Answer: The head might show on the first toss, or on the second toss, or on the third toss, continuing indefinitely. The first head could appear after one toss (H), two tails then a head (TH), three tails then a head (TTH), four tails then a head (TTTH), and so on. Therefore, the sample space is S = {H, TH, TTH, TTTH, TTTTH, ...}. Since the head could theoretically appear on any toss from 1 onwards, the number of elements in this sample space is infinite.
In simple words: Keep tossing until heads appears. Heads could come on the first try, or after many tails - there is no limit to when it might appear.

Exam Tip: Recognize that some experiments have infinite sample spaces - these occur when the process continues indefinitely until a certain condition is met, like "first success" scenarios.

Example 10. We wish to choose one child out of 2 boys and 3 girls. A coin is tossed. If it comes up heads, a boy is chosen, otherwise a girl is chosen. Describe the sample space.
Answer: The coin outcome can be heads (H) or tails (T). Label the boys B1 and B2, and the girls G1, G2, and G3. If heads appears, one of the two boys is chosen. If tails appears, one of the three girls is chosen. The sample space is S = {HB1, HB2, TG1, TG2, TG3}.
In simple words: The coin decides the gender (heads means boy, tails means girl), then a specific person of that gender is selected.

Exam Tip: When combining independent selections (coin toss to decide category, then pick a specific person), list all combinations systematically to form the complete sample space.

Exercise 16.1 - Very Short Answer Type Questions (1 to 3)

Question 1. Describe the sample space for the following experiments:
(i) Two coins are tossed.
(ii) One coin is tossed twice.
(iii) A coin is tossed three times.
(iv) Three coins are tossed simultaneously.
(v) A die is rolled twice.
(vi) A coin is tossed and a die is thrown.
(vii) A coin is tossed, and if head comes up, a die is thrown.
(viii) A coin is tossed thrice and number of heads is recorded.
(ix) A card is drawn from a deck of playing cards, and its colour is noted.
(x) A card is drawn from a deck of playing cards, and its suit is noted.

Answer:

(i) {HH, HT, TH, TT}

(ii) {HH, HT, TH, TT}

(iii) {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}

(iv) {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}

(v) {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}

(vi) {H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}

(vii) {H1, H2, H3, H4, H5, H6, T}

(viii) {0, 1, 2, 3}

(ix) {Red, Black}

(x) {Spade, Heart, Diamond, Club}
In simple words: Each experiment's sample space lists all distinct outcomes that can happen.

Exam Tip: Count carefully to ensure your list is complete - for identical experiments (like two coins tossed or one coin tossed twice), the sample space is the same, but when recording information differs (like counting heads instead of listing sequences), the space changes.

Question 2. A box contains 1 white and 3 identical green balls. Two balls are drawn one by one without replacement. Write the sample space for this experiment.
Answer: Denote the white ball as W and the green balls as G1, G2, G3 (even though they are identical, we label them for clarity in listing outcomes). Drawing two balls one by one without replacement, the possible outcomes are: (W, G1), (W, G2), (W, G3), (G1, W), (G2, W), (G3, W), (G1, G2), (G1, G3), (G2, G3), (G2, G1), (G3, G1), (G3, G2). Since the green balls are identical, we can also represent the sample space more simply as {WG, GW, GG}.
In simple words: You can pull either the white ball or a green ball first, then pull the remaining ball second.

Exam Tip: When drawing without replacement, the second draw depends on what was drawn first - account for this by ensuring no ball is used twice in an outcome.

Question 3. Consider the experiment in which a coin is tossed repeatedly until a tail comes up for the first time. Describe the sample space.
Answer: The tail could appear on the first toss (outcome T), or after heads on the first and tail on the second (outcome HT), or after heads on first two tosses and tail on third (outcome HHT), or after heads on first three tosses and tail on fourth (outcome HHHT), and so on. The experiment continues until a tail shows up. The sample space is S = {T, HT, HHT, HHHT, HHHHT, ...}, which contains infinitely many elements.
In simple words: Toss until tails appears - it could happen on try 1, try 2, try 3, or any number of tries.

Exam Tip: Recognize that "until" problems create infinite sample spaces - there is no upper limit on when the stopping condition (tail in this case) occurs.

Question 4. In team A, there are 2 boys and 2 girls. In team B, there is one boy and 3 girls. First a team is chosen, and then a participant. Describe the sample space.
Answer: Team A has boys b1, b2 and girls g1, g2. Team B has boy b3 and girls g3, g4, g5. The process involves two steps: choose a team, then choose a person from that team. If team A is chosen, we can pick any of the 4 members: Ab1, Ab2, Ag1, Ag2. If team B is chosen, we can pick any of the 4 members: Bb3, Bg3, Bg4, Bg5. The sample space is S = {Ab1, Ab2, Ag1, Ag2, Bb3, Bg3, Bg4, Bg5}.
In simple words: First, pick which team. Then, pick a person from that team.

Exam Tip: For two-stage selections like this, complete all choices at each stage systematically - choose team A fully before moving to team B.

Question 5. A bag contains 2 red and 3 black balls. What is the sample space when the experiment consists of drawing?
(i) 1 ball
(ii) 2 balls (assuming that order is not important)

Answer:

(i) If drawing 1 ball, label red balls as r1, r2 and black balls as b1, b2, b3. The sample space is S = {r1, r2, b1, b2, b3}.

(ii) If drawing 2 balls without order mattering (combinations), the possible outcomes are: {r1, r2}, {r1, b1}, {r1, b2}, {r1, b3}, {r2, b1}, {r2, b2}, {r2, b3}, {b1, b2}, {b1, b3}, {b2, b3}. The sample space is S = {r1r2, r1b1, r1b2, r1b3, r2b1, r2b2, r2b3, b1b2, b1b3, b2b3}.
In simple words: Pulling one ball gives five choices. Pulling two balls without considering order gives ten possible combinations.

Exam Tip: When order does not matter, use combinations (groupings) rather than ordered pairs - ensure you list each unique pair only once.

16.2 Events

Event

A subset of the sample space of a random experiment is referred to as an event. When a die is thrown, the sample space is S = {1, 2, 3, 4, 5, 6}. Various events can be described based on how the outcome is reported: {6} represents getting a six; {2, 4, 6} represents getting an even number; {1, 3, 5} represents getting an odd number; {2, 3, 5} represents getting a prime number; {1, 2, 3, 4} represents getting a number less than 5. All of these are events within this experiment. Similarly, when two coins are tossed, the sample space is S = {HH, HT, TH, TT}. Several events can occur: {HH} means getting two heads; {HT, TH} means getting exactly one head; {TT} means getting no heads; {HH, HT, TH} means getting at least one head.

16.2.1 Occurrence of an Event

An event is said to have occurred when the actual result of an experiment matches the condition stated in the event definition. For example, in throwing a die, if event E is "obtaining an even number," then event E is considered to have happened if the die shows 2, 4, or 6. If the die shows 1, 3, or 5, then event E has not occurred. Similarly, when tossing two coins, if event E is "obtaining two heads," then E occurs only if both coins show heads; otherwise, E does not occur.

16.2.2 Types of Events

Simple Event (or Elementary Event): A simple event is one specific possible outcome of an experiment. For instance, when 3 coins are tossed, {HHH} and {HTH} are each simple events. A simple event contains exactly one sample point.

Compound Event (or Decomposible Event): A compound event occurs when two or more simple events happen together. In other words, if an event contains more than one sample point from the sample space, it is a compound event. For example, when 3 coins are tossed, {HHT, HTH, THH, HHH} is a compound event corresponding to "obtaining at least two heads." This event groups multiple simple events together.

Sure Event: A sure event is the entire sample space S itself. When throwing a die, S = {1, 2, 3, 4, 5, 6}. The event "obtaining a number less than 7" equals the sample space and is therefore a sure event because every outcome of the experiment satisfies this condition. In contrast, "obtaining a number less than 5" is not a sure event since some outcomes (5 and 6) do not satisfy it.

Impossible Event: An impossible event corresponds to the empty set ∅. It represents an outcome that cannot happen in the experiment. For example, when rolling a die, "obtaining a number greater than 6" is an impossible event because no die face shows a number exceeding 6.

Equally Likely Outcomes: Outcomes are said to be equally likely when there is no logical reason to favor one outcome over another. Each outcome has the same probability of occurring. For example, when drawing a card from a well-shuffled deck of 52 cards, all 52 outcomes are equally likely. Getting a spade is just as likely as getting a heart (both have 13 cards); getting a king is just as likely as getting a queen (both have 4 cards).

16.2.3 Algebra of Events

Since events can be represented as sets, the standard set operations - including union, intersection, complement, and difference - can be applied to events just as they are applied to sets.

Complement of an Event

The complement of an event E, denoted by Ē, E′, or E^c, refers to the set containing all outcomes in the sample space S except those in E. Formally, E^c = {w : w ∈ S and w ∉ E} = S - E. It follows that E ∪ E^c = S and E ∩ E^c = ∅. Consider two coins being tossed, where the sample space is S = {HH, HT, TH, TT}. If E represents the event "at least one tail appears" = {HT, TH, TT}, then E^c or "not E" = {HH} = {no tail appears}. The complement captures all outcomes where the original event condition is not met.

The Event "A or B"

The union of two sets A and B, written as A ∪ B and called "A or B," is the set containing all elements that belong to A, or to B, or to both. For events, "A or B" is defined as A ∪ B = {w : w ∈ A or w ∈ B}. This event occurs whenever at least one of the two component events happens.

The Event "A and B"

The intersection of two sets A and B, denoted by A ∩ B and called "A and B," is the set of all elements that belong to both A and B. For events, "A and B" is A ∩ B = {w : w ∈ A and w ∈ B}. This event occurs only when both component events happen simultaneously.

The Event "A but not B"

The difference of two sets A and B, written as A - B, consists of all elements that are in A but not in B. For events, "A but not B" is A - B = {w : w ∈ A and w ∉ B} = A ∩ B′. This event happens when A occurs but B does not occur.

Exhaustive Events

Events E1, E2, ..., En associated with a random experiment and sample space S are called exhaustive if E1 ∪ E2 ∪ ... ∪ En = S. This means the union of all these events covers the entire sample space. For example, if S = {1, 2, 3, 4, 5, 6}, A = {1, 3, 5}, B = {2, 4, 6}, and C = {1, 2, 3, 4}, then A ∪ B = S, so A and B are exhaustive events. However, A ∪ C = {1, 2, 3, 4, 5} ≠ S, so A and C are not exhaustive. A key point: when events are exhaustive, at least one of them must necessarily occur every time the experiment is performed.

Mutually Exclusive Events

Two or more events associated with a random experiment are called mutually exclusive if the happening of any one event prevents the occurrence of all other events. If A and B are mutually exclusive, then A ∩ B = ∅ (the empty set). In the example above, A ∩ B = {1, 3, 5} ∩ {2, 4, 6} = ∅, so A and B are mutually exclusive. However, A ∩ C = {1, 3, 5} ∩ {1, 2, 3, 4} = {1, 3} ≠ ∅, so A and C are not mutually exclusive. Similarly, in a deck of cards, let A = {getting a spade}, B = {getting a heart}, C = {getting a king}. Then A and B are mutually exclusive since A ∩ B = ∅, whereas A and C are not mutually exclusive because A ∩ C = {king of spade} ≠ ∅. Note that the simple events of a sample space are always mutually exclusive.

Mutually Exclusive and Exhaustive Events

Let S be the sample space for a random experiment. Events E1, E2, ..., En of S are called mutually exclusive and exhaustive if E1 ∪ E2 ∪ ... ∪ En = S (they are exhaustive) AND Ei ∩ Ej = ∅ for all i ≠ j (they are mutually exclusive - no two events overlap). For example, if A = {getting a spade}, B = {getting a club}, C = {getting a heart}, D = {getting a diamond}, then A, B, C, D are mutually exclusive and exhaustive events because these four sets cover all 52 cards with no overlap.

Exhaustive Number of Cases

The exhaustive number of cases represents the total count of all possible outcomes when an experiment is performed. When tossing a coin, the exhaustive number of cases is 2 (heads or tails). When throwing a die, it is 6. When throwing three dice simultaneously, it is 6³ = 216. When drawing 2 cards simultaneously from a 52-card deck, it is ₅₂C₂.

Favorable Number of Cases

The favorable number of cases for an event is the count of outcomes that satisfy the event's condition. For instance, when three coins are tossed, the favorable number of cases for the event "getting at least two heads" is 4, corresponding to outcomes HHH, HHT, HTH, THH. Can you determine the favorable number of cases for the event "getting a sum of 10" when two dice are thrown simultaneously?

Example 1. (i) From a group of 3 boys and 2 girls, we select two children. Describe the sample space. How would you represent the events "both selected children are boys" and "one boy and one girl is selected"? (ii) A coin is tossed. If it shows head, we draw a ball from a bag consisting of 3 red and 4 black balls; if it shows tails, we throw a die. What is the sample space of this experiment? How would you represent the events "coin came up head" and "the throw of the die resulted in an even number"?
Answer:

(i) Selecting two children from five can be done in ₅C₂ = 10 ways. If boys are labeled B1, B2, B3 and girls are labeled G1, G2, the sample space is S = {B1B2, B1B3, B1G1, B1G2, B2B3, B2G1, B2G2, B3G1, B3G2, G1G2}. The event "both selected children are boys" consists of outcomes where both are boys: E1 = {B1B2, B1B3, B2B3}. The event "one boy and one girl is selected" consists of outcomes with one of each: E2 = {B1G1, B1G2, B2G1, B2G2, B3G1, B3G2}.

(ii) Label red balls r1, r2, r3 and black balls b1, b2, b3, b4. The sample space is S = {Hr1, Hr2, Hr3, Hb1, Hb2, Hb3, Hb4, T1, T2, T3, T4, T5, T6}. The event "coin came up head" includes all outcomes starting with H: E1 = {Hr1, Hr2, Hr3, Hb1, Hb2, Hb3, Hb4}. The event "the throw of the die resulted in even number" includes die results 2, 4, 6: E2 = {T2, T4, T6}.
In simple words: Events are subsets of the sample space - they collect all the outcomes that meet the specific condition.

Exam Tip: Clearly state the condition for each event, then list all outcomes that satisfy that condition - ensure no relevant outcome is left out.

Example 2. A die is thrown twice. Describe the sample space of this experiment. Let E1 = {both numbers are even}, E2 = {both numbers are odd}, E3 = {sum is less than 6}. Describe E1, E2, E3, E1 ∪ E2, E1 ∩ E2, E1 ∪ E3, E1 ∩ E3, E1^c, E2^c.
Answer: The sample space is S = {(x, y); x, y ∈ {1, 2, 3, 4, 5, 6}} containing all 36 ordered pairs.

E1 = {both numbers are even} = {(2, 2), (2, 4), (2, 6), (4, 2), (4, 4), (4, 6), (6, 2), (6, 4), (6, 6)}

E2 = {both numbers are odd} = {(1, 1), (1, 3), (1, 5), (3, 1), (3, 3), (3, 5), (5, 1), (5, 3), (5, 5)}

E3 = {sum is less than 6} = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (4, 1)}

E1 ∪ E2 = {(2, 2), (2, 4), (2, 6), (4, 2), (4, 4), (4, 6), (6, 2), (6, 4), (6, 6), (1, 1), (1, 3), (1, 5), (3, 1), (3, 3), (3, 5), (5, 1), (5, 3), (5, 5)}

E1 ∩ E2 = ∅ (no outcome has all even numbers AND all odd numbers)

E1 ∪ E3 = {(2, 2), (2, 4), (2, 6), (4, 2), (4, 4), (4, 6), (6, 2), (6, 4), (6, 6), (1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 3), (3, 1), (3, 2), (4, 1)}

E1 ∩ E3 = {(2, 2)} (the only outcome where both are even AND the sum is less than 6)

E1^c = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 3), (2, 5), (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (4, 1), (4, 3), (4, 5), (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), (6, 1), (6, 3), (6, 5)} (all pairs where at least one number is odd)

E2^c = {(1, 2), (1, 4), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 2), (3, 4), (3, 6), (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), (5, 2), (5, 4), (5, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)} (all pairs where at least one number is even)
In simple words: Union combines outcomes from both events. Intersection keeps only shared outcomes. Complement flips the condition - everything NOT in the original event.

Exam Tip: When finding unions, intersections, and complements, carefully check each outcome against the event conditions before including or excluding it.

Example 3. Two dice are rolled. A is the event that the sum of the numbers shown on the two dice is 5. B is the event that atleast one of the dice shows up a 3. Are the two events A and B (i) mutually exclusive (ii) exhaustive? Give arguments in support of your answer.
Answer: The sample space has 36 outcomes, so n(S) = 36.

A = {(1, 4), (2, 3), (3, 2), (4, 1)} (all pairs summing to 5)

B = {(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (1, 3), (2, 3), (4, 3), (5, 3), (6, 3)} (all pairs with at least one 3)

(i) To check mutual exclusivity: A ∩ B = {(2, 3), (3, 2)} ≠ ∅. Since the intersection is not empty, A and B are not mutually exclusive - they share two common outcomes.

(ii) To check exhaustiveness: A ∪ B = {(1, 4), (2, 3), (3, 2), (4, 1), (3, 1), (3, 3), (3, 4), (3, 5), (3, 6), (1, 3), (4, 3), (5, 3), (6, 3)}, so n(A ∪ B) = 13. Since 13 < 36, we have A ∪ B ≠ S. Therefore, A and B are not exhaustive events.
In simple words: Events are mutually exclusive when they cannot happen together (no overlap). They are exhaustive when together they cover all possibilities.

Exam Tip: For mutual exclusivity, find the intersection - if it is empty, they are mutually exclusive. For exhaustiveness, check if the union equals the entire sample space.

Example 4. A coin is tossed three times. Consider the following events: A: 'No head appears', B: 'exactly one head appears', C: 'Atleast two heads appear'. Do they form a set of mutually exclusive and exhaustive events?
Answer: The sample space is S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}.

A = {TTT} (no heads)

B = {HTT, THT, TTH} (exactly one head)

C = {HHT, HTH, THH, HHH} (at least two heads)

Check exhaustiveness: A ∪ B ∪ C = {TTT, HTT, THT, TTH, HHT, HTH, THH, HHH} = S. All events together cover the entire sample space, so they are exhaustive.

Check mutual exclusivity: A ∩ B = ∅, A ∩ C = ∅, B ∩ C = ∅. No two events share any outcomes.

Therefore, A, B, and C form a set of mutually exclusive and exhaustive events.
In simple words: These three categories (0 heads, 1 head, 2+ heads) cover every possible outcome with no overlap between categories.

Exam Tip: When checking if events are mutually exclusive and exhaustive, verify that their union equals S and that all pairwise intersections are empty.