ML Aggarwal Class 12 Maths Solutions Section A Chapter 13 Probability

13 Probability

Introduction

Statements like "It may rain today," "Rajesh is quite sure to top his class," "It is highly unlikely that Salman will marry Preeti," and "I have bet 100 rupees on India winning against Pakistan" all show examples of uncertainty in our lives. Probability gives us a way to measure how likely something is to happen. The study of probability started when mathematicians looked at games where people gamble and try to win money. For example, imagine you pay Rs. 2 to pull a card from a pack of 52 cards. If you get an ace, you receive 20 rupees; otherwise you get nothing. Should you play this game? Will you have a fair chance to win, or will you lose money over time? Questions like these led to the growth of probability theory. Italian mathematician Jerome Cardan, French mathematicians Pascal and Pierre de Fermat, and Swiss mathematician James Bernouli started this field of study.

13.1 Random Experiments and Sample Spaces

An Experiment

An action or operation that produces outcomes with clear results is called an experiment. Some experiments always give the same result. For instance, if you are given any triangle and don't know the three angles, you can still be certain that those angles add up to 180°. We call these deterministic experiments. Other experiments may give different results. For example, when you flip a coin, you might get heads or tails. These are called probabilistic or random experiments.

Random Experiment

An experiment is random when it can produce two or more different results and you cannot know in advance which result will happen. Examples 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 pack of 52 playing cards. From this point on, whenever we say "experiment," we mean a random experiment.

Sample Space

The sample space is the group of all possible results from a random experiment. We usually write it as S. For instance, when you toss a coin once, there are two possible outcomes - heads or tails. If we show heads as H and tails as T, then the sample space is S = {H, T}. When you roll a die, it can land with any of its 6 faces up, giving us any of the six numbers 1, 2, 3, 4, 5, 6. So the sample space for rolling a die is S = {1, 2, 3, 4, 5, 6}.

Illustrative Examples

Example 1. Describe the sample space when two coins are tossed together.
Answer: We can use ordered pairs to list all results. The sample space is S = {(H, H), (H, T), (T, H), (T, T)}, where (H, T) means the first coin shows heads and the second shows tails, and so on.
In simple words: When you toss two coins, you can get heads-heads, heads-tails, tails-heads, or tails-tails. These four results make up the sample space.

Exam Tip: Always use ordered pairs when order matters (e.g., first coin and second coin are different). Remember the total number of outcomes is the product of outcomes for each 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 results. We can write it as 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)}. We can also write it more simply as S = {H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}.
In simple words: When you flip a coin and roll a die at the same time, you get one of two coin results (heads or tails) paired with one of six die results (1 through 6). That makes 12 possible results altogether.

Exam Tip: When combining independent experiments, multiply the number of outcomes from each to get the total number of outcomes in the combined experiment.

Example 3. Find the sample space associated with the experiment of rolling a pair of dice once. Also find the number of elements of the sample space.
Answer: The first die can show any number from 1 to 6, and the second die can also show any number from 1 to 6. We can describe each result as an ordered pair (x, y), where x is the number on the first die and y is the number on the second die. The sample space is given by S = {(x, y) : x, y ∈ {1, 2, 3, 4, 5, 6}}. We can list all results in a table:

Exam Tip: Present the sample space clearly using either set notation or a table for clarity. Always count correctly - for two dice, multiply 6 × 6 to get 36 total outcomes.

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) The elder child can be either a boy or a girl. So there are only two possible outcomes: S = {Boy, Girl} or S = {B, G}.
(ii) Since there are 2 children, there are 4 possible outcomes - boy-boy, boy-girl, girl-boy, girl-girl. The sample space is S = {BB, BG, GB, GG}.
(iii) Since we only care about how many girls there are, there are 3 possible outcomes - zero girls, one girl, or two girls. So the sample space is S = {0, 1, 2}.
In simple words: The sample space changes depending on what we want to know. If we only look at the older child, there are 2 outcomes. If we care about birth order, there are 4 outcomes. If we only count girls, there are 3 outcomes.

Exam Tip: Always identify what information the question asks for - this determines which sample space to use. Different questions about the same experiment can have different sample spaces.

Example 5. 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: We can show all the paths using a tree diagram. The experiment works like this: we flip a coin twice. If we get heads both times (HH), we stop. If the first flip is heads but the second is tails (HT), we roll a die. Similarly, if the first flip is tails and the second is heads (TH), we stop. But if we get tails on the second flip (whether the first was heads or tails), we roll a die. This creates different paths through the experiment. 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: When conditions change what happens next in an experiment, we get different branches. Some paths stop early (like HH and TH), while others continue with the die roll (HT and TT followed by numbers 1-6).

Exam Tip: For experiments where one action depends on the result of another, always draw a tree diagram to track all possible paths. Count all endpoints to get the total number of outcomes.

Example 6. 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: Let's follow what happens. First, we flip a coin. If we get tails (T), the experiment ends. If we get heads (H), we roll a die. If the die shows 2, 4, or 6 (even numbers), the experiment stops. If it shows 1 or 3, we roll the die again. If it shows 5, we flip a coin. Using a tree diagram to track all paths: The sample space is S = {T, H2, H4, H6, H11, H12, H13, H14, H15, H16, H31, H32, H33, H34, H35, H36, H5H, H5T}. There are 18 possible outcomes.
In simple words: This is a multi-stage experiment where the outcome of one stage decides what happens next. We must trace through all branches to find every possible ending.

Exam Tip: For complex multi-stage experiments, tree diagrams are essential for not missing any path. Label each branch clearly and count every terminal point.

Example 7. 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: We keep flipping the coin until heads appears. Heads might appear on the first flip, or on the second flip after one tail, or on the third flip after two tails, and so on. The sample space is S = {H, TH, TTH, TTTH, TTTTH, ...}. The number of outcomes in this sample space is infinite, because we could theoretically get any number of tails before finally getting heads.
In simple words: This experiment can go on forever since we might keep getting tails for many flips before finally getting heads. That's why we have infinitely many possible outcomes.

Exam Tip: Some experiments can theoretically continue indefinitely, leading to an infinite sample space. Represent such spaces using set notation with "..." to show the pattern continuing.

Example 8. 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 flip result can be heads (H) or tails (T). Let's name the boys B1 and B2, and the girls G1, G2, and G3. If the coin shows heads, we pick a boy. If it shows tails, we pick a girl. So the sample space is S = {HB1, HB2, TG1, TG2, TG3}. Each outcome pairs the coin result with a specific child.
In simple words: This combines a coin flip with a selection. If heads, we get one of two boys. If tails, we get one of three girls. That's 5 total results.

Exam Tip: When combining separate events (like coin flip and selection), list all pairs carefully. The total outcomes equal the product of individual outcomes when combined.

Exercise 13.1

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.

2. 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.

3. 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)

4. Redo the above problem assuming that the 2 red balls are identical and the three black balls are identical. Assume that after drawing one ball, it is replaced before drawing the second ball.

5. A coin is tossed. If it shows head, we draw a ball from a bag consisting of 2 blue and 3 white balls; if it shows tail, we toss the coin again. Describe the sample space.

6. Consider the experiment in which a coin is tossed repeatedly until a tail comes up for the first time. Describe the sample space.

7. One die of red colour, one of white colour and one of blue colour are placed in a bag. One die is selected at random and rolled, its colour and the number on its uppermost face is noted. Describe the sample space.

8. Suppose 3 bulbs are selected at random from a lot. Each bulb is tested and classified as defective (D) or non-defective (N). Write the sample space of this experiment.

9. The numbers 1, 2, 3 and 4 are written separately on four slips of paper. The slips are then put in a box and mixed thoroughly. A person draws two slips from the box, one after the other, without replacement. Describe the sample space of the experiment.

10. A box contains 1 red and 3 identical white balls. Two balls are drawn at random in succession without replacement. Write the sample space for this experiment.

11. An experiment consists of rolling a die and then tossing a coin once if the number on the die is even. If the number on the die is odd, the coin is tossed twice. Draw the tree diagram for the possible outcomes. Write the sample space for this experiment. How many outcomes are possible in this experiment?

13.2 Events

Event

Any group that is part of the sample space from a random experiment is called an event. For example, when a die is rolled, the sample space is S = {1, 2, 3, 4, 5, 6}. We can describe the result in many ways: rolling a 6 corresponds to {6}, rolling an even number corresponds to {2, 4, 6}, rolling an odd number corresponds to {1, 3, 5}, rolling a prime number corresponds to {2, 3, 5}, rolling a number less than 5 corresponds to {1, 2, 3, 4}, and so on. Each of these is an event. When two coins are tossed, the sample space is S = {HH, HT, TH, TT}. Some events for this experiment are: getting exactly two heads - {HH}, getting exactly one head - {HT, TH}, getting no heads - {TT}, getting at least one head - {HH, HT, TH}, and so on.

Occurrence of Event

When a result from an experiment fits what the event describes, we say that the event has happened. For example, if we roll a die and define event E as "getting an even number," then whenever the die shows 2, 4, or 6, we say event E occurred. If it shows 1, 3, or 5, we say event E did not occur. In another case, if we toss two coins and event E is "getting two heads," then event E occurs only if both coins show heads.

13.2.1 Types of Events

Simple Event

A simple event is one specific possible outcome from an experiment. When 3 coins are tossed, results like {HHH} and {HTH} are simple events. We also call a simple event an elementary event or indecomposible event - it cannot be broken down into smaller parts.

Compound Event

A compound event happens when two or more simple events occur together. Another way to say this is that if an event contains more than one outcome point, it is a compound event. When 3 coins are tossed, the set {HHT, HTH, THH, HHH} is a compound event because it represents "getting a minimum of two heads." We also call a compound event a decomposible event because it can be separated into simpler parts.

Sure Event

A sure event is the sample space S itself - an event that will definitely happen. When we roll a die, S = {1, 2, 3, 4, 5, 6}. The event "getting a number less than 7" is a sure event because every result satisfies this. However, "getting a number less than 5" is not a sure event because some outcomes (5 and 6) do not satisfy this.

Impossible Event

An impossible event matches the empty set φ - it can never happen. When rolling a die, the event "getting a number higher than 6" is impossible because no die face shows a number larger than 6.

Equally Likely Outcomes

We say outcomes are equally likely when we have no reason to believe one outcome is more probable than another. When we draw a card from a well-shuffled deck of 52 cards, all 52 possible outcomes have an equal chance of happening. Drawing a spade is just as likely as drawing a heart (since there are 13 of each). Getting a king is just as likely as getting a queen (since there are 4 of each).

13.2.2 Algebra of Events

Since we can think of events as sets, we can use the same operations we use with sets - like finding unions, intersections, complements, and differences - on events too.

Complement of an Event

The complement of event E, shown as E' or E^c or Ē, is the collection of all outcomes in the sample space that are NOT in E. In other words, E^c = {w : w ∈ S and w ∉ E} = S - E. Note that E ∪ E^c = S (together they make the whole sample space) and E ∩ E^c = φ (they share no outcomes). For example, when two coins are tossed, the sample space is S = {HH, HT, TH, TT}. If E is the event "at least one tail appears" = {HT, TH, TT}, then E^c or "not E" = {HH} = {no tail appears}.

The Event "A or B"

Recall that the union of two sets A and B, written A ∪ B and called "A or B," includes all elements that are in A, or in B, or in both. Similarly, event "A or B" means A ∪ B = {w : w ∈ A or w ∈ B}.

The Event "A and B"

The intersection of two sets A and B, written A ∩ B and called "A and B," includes all elements that belong to both A and B. Similarly, event "A and B" means A ∩ B = {w : w ∈ A and w ∈ B}.

The Event "A but not B"

The difference between sets A and B, written A - B, includes elements that are in A but not in B. Similarly, event "A but not B" means A - B = {w : w ∈ A and w ∉ B} = A ∩ B'.

Exhaustive Events

Events E₁, E₂, ..., Eₙ linked with a random experiment are called exhaustive when E₁ ∪ E₂ ∪ ... ∪ Eₙ = S. For example, if S = {1, 2, 3, 4, 5, 6}, A = {1, 3, 5}, and B = {2, 4, 6}, then A ∪ B = S, so A and B are exhaustive events. But if C = {1, 2, 3, 4}, then A ∪ C = {1, 2, 3, 4, 5} ≠ S, so A and C are not exhaustive. When events are exhaustive, at least one of them must definitely happen when the experiment is performed.

Mutually Exclusive Events

Two or more events from a random experiment are called mutually exclusive when if one occurs, the others cannot occur. If A and B are mutually exclusive, then A ∩ B = φ (the empty set, meaning they have nothing in common). In the earlier example with S = {1, 2, 3, 4, 5, 6}, A = {1, 3, 5}, and B = {2, 4, 6}, we have A ∩ B = {1, 3, 5} ∩ {2, 4, 6} = φ. So A and B are mutually exclusive. However, if C = {1, 2, 3, 4}, then A ∩ C = {1, 3, 5} ∩ {1, 2, 3, 4} = {1, 3} ≠ φ. So A and C are not mutually exclusive. For playing cards, if A = {getting a spade}, B = {getting a heart}, and C = {getting a king}, then A and B are mutually exclusive (because A ∩ B = φ), but A and C are not mutually exclusive (because A ∩ C = {king of spade} ≠ φ). Remark: The simple events of a sample space are always mutually exclusive.

Mutually Exclusive and Exhaustive Events

When S is the sample space from a random experiment, the events E₁, E₂, ..., Eₙ of S are called mutually exclusive and exhaustive when E₁ ∪ E₂ ∪ ... ∪ Eₙ = S (meaning they cover everything) and Eᵢ ∩ Eⱼ = φ for all i ≠ j (meaning no two overlap). For example, if A = {getting a spade}, B = {getting a club}, C = {getting a heart}, and D = {getting a diamond}, then A, B, C, and D are mutually exclusive and exhaustive events.

Exhaustive Number of Cases

The exhaustive number of cases is the total count of all possible outcomes from an experiment. When a coin is tossed, the exhaustive number of cases is 2 (heads or tails). When a die is rolled, the exhaustive number of cases is 6. When three dice are rolled simultaneously, the exhaustive number of cases is 6³ = 216. When 2 cards are drawn at the same time from a deck, the exhaustive number of cases is 52C₂.

Favourable Number of Cases

The favourable number of cases is how many results support (or are favourable to) an event. For instance, when three coins are tossed, the event "getting a minimum of two heads" is satisfied by the cases HHH, HHT, HTH, and THH. So there are 4 favourable cases for this event. As a challenge: Can you find how many favourable cases there are for the event "getting a sum of 10" when two dice are rolled at the same time?

Illustrative Examples

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) We can pick two children from five in 5C₂ = 10 ways. If we call the boys B₁, B₂, B₃ and the girls G₁, G₂, the sample space is S = {B₁B₂, B₁B₃, B₁G₁, B₁G₂, B₂B₃, B₂G₁, B₂G₂, B₃G₁, B₃G₂, G₁G₂}. The event "both selected children are boys" is E₁ = {B₁B₂, B₁B₃, B₂B₃}. The event "one boy and one girl is selected" is E₂ = {B₁G₁, B₁G₂, B₂G₁, B₂G₂, B₃G₁, B₃G₂}.
(ii) If we label the red balls as r₁, r₂, r₃ and the black balls as 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" is E₁ = {Hr₁, Hr₂, Hr₃, Hb₁, Hb₂, Hb₃, Hb₄}. The event "the throw of the die resulted in even number" is E₂ = {T2, T4, T6}.
In simple words: Part (i): We list all ways to pick 2 children from 5, then group them by what happened. Part (ii): We track what happened - either we picked a ball (if heads) or rolled a number (if tails).

Exam Tip: Always define events by clearly listing which outcomes they include. Use clear notation for sample points to avoid confusion.

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, E₁ᶜ, E₂ᶜ.
Answer: The sample space is S = {(x, y) : x, y ∈ {1, 2, 3, 4, 5, 6}} or S = {(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)}. 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 = {both numbers are even or both numbers are odd} = {(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 = {both numbers are even and both numbers are odd} = φ (the empty set, because a number cannot be both even and odd) E₁ ∪ E₃ = {both numbers are even or sum is less than 6} = {(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 = {both numbers are even and sum is less than 6} = {(2,2)} E₁ᶜ = {outcomes except where both numbers are even} = {(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)} E₂ᶜ = {outcomes except where both numbers are odd} = {(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)}
In simple words: E₁ includes pairs where both numbers are even. E₂ includes pairs where both are odd. E₃ includes pairs that add up to less than 6. When we combine or overlap events, we list the results accordingly.

Exam Tip: When finding unions, list all outcomes from both events. When finding intersections, list only outcomes that appear in both. For complements, include everything in the sample space except that event.

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 S has 36 points, so n(S) = 36. Event A (sum is 5) = {(1,4), (2,3), (3,2), (4,1)} and Event B (at least one die shows 3) = {(3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (1,3), (2,3), (4,3), (5,3), (6,3)}. Since A ∩ B = {(2,3), (3,2)} ≠ φ, events A and B are not mutually exclusive - they share two outcomes. Also, n(A ∪ B) = 13 < n(S), meaning not all 36 outcomes are covered. So A and B are not exhaustive - there are outcomes not in either event, like (1,1) or (6,6).
In simple words: Events are mutually exclusive if they cannot both happen at once. Here they can - rolling (2,3) gives both a sum of 5 and a 3 on one die. Events are exhaustive if at least one must happen. Here many outcomes (like getting a sum of 8 with no 3) belong to neither.

Exam Tip: Check mutual exclusivity by finding A ∩ B - if it is empty, they are mutually exclusive. Check exhaustiveness by checking if A ∪ B equals the entire sample space - if not, they are not exhaustive.

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}. Now A = {TTT}, B = {HTT, THT, TTH}, C = {HHT, HTH, THH, HHH}. First, A ∪ B ∪ C = {TTT, HTT, THT, TTH, HHT, HTH, THH, HHH} = S. This shows the events are exhaustive - they cover all possibilities. Second, A ∩ B = φ (no overlap between A and B), A ∩ C = φ (no overlap between A and C), and B ∩ C = φ (no overlap between B and C). This shows the events are mutually exclusive - no outcome belongs to more than one event. Therefore, A, B, and C form a set of mutually exclusive and exhaustive events.
In simple words: These three events divide all possible outcomes into distinct groups. Every possible flip result falls into exactly one group - either no heads, exactly one head, or at least two heads. There is no overlap.

Exam Tip: To verify mutually exclusive and exhaustive: (1) Check that each pair of events has an empty intersection, (2) Check that the union of all events equals the entire sample space.

Exercise 13.2

1. (i) A coin is tossed once. Write its sample space and all possible events.
(ii) A coin is tossed twice. Write its sample space and all possible events.
(iii) A coin is tossed n times. What is the number of elements in its sample space? Also write the number of all possible events.
Hint. Let S be the sample space. As a sequence of n items XX...X occurs when 'X' may either be H or T, so there are 2ⁿ possible outcomes. Hence O(S) = 2ⁿ. Number of possible events = number of subsets of S = 2^O(S).

2. If three dice are thrown together, how many outcomes would the sample space have? Represent the event when all dice come up with same number.

3. A coin and a die are tossed. Describe the sample space and the following events.
(i) A = getting a head and an even number.
(ii) B = getting a prime number.
(iii) C = getting a tail and an odd number.
(iv) D = getting a head or a tail.
Now answer the following questions.
(a) Are A and B mutually exclusive?
(b) Are A and C mutually exclusive?
(c) Are A and C exhaustive?
(d) Are A and D exhaustive?

4. Three coins are tossed.
(i) Describe two events A and B which are mutually exclusive.
(ii) Describe three events A, B and C which are mutually exclusive and exhaustive.
(iii) Describe two events which are not mutually exclusive.
(iv) Describe three events which are not mutually exclusive.
(v) Describe two events which are mutually exclusive but not exhaustive.
(vi) Describe three events which are mutually exclusive but not exhaustive.

5. Two dice are thrown. If E is the event "both dice come up with same number" and F is the event "product of the numbers on the two dice is odd", then describe
(i) E
(ii) F
(iii) E or F
(iv) E and F
(v) E but not F

6. A pair of dice is rolled. Consider the following events
A: the sum is greater than 8
B: 2 occurs on either die
C: the sum is at least 7 and a multiple of 3.
(i) Find A, B, C, A ∩ B, B ∩ C and A ∩ C.
(ii) Which pairs of events are mutually exclusive?

7. From a group of 2 men and 3 women, two persons are selected. Describe the sample space of the experiment. If E is the event in which one man and one woman are selected, then which are the cases favourable to E?

8. A coin is tossed thrice. If E denotes the 'number of heads is odd' and F denotes the 'number of tails is odd,' then find the cases favourable to the event E ∩ F.

9. If E, F, G denote the subsets representing the events of a sample space S, what are the sets representing the following events?
(i) Out of the three events at least two events occur.
(ii) Out of the three events only one occurs.
(iii) Out of the three events only E occurs.
(iv) Out of the three events exactly two events occur.

10. Two dice are rolled. Let A, B and C be the events of getting a sum 2, sum 3, and a sum 4, respectively.
(i) Is event A simple?
(ii) Is event B simple?
(iii) Is event C compound?
(iv) Are events A and B mutually exclusive?

11. Three coins are tossed once. Let A denote the event "three heads show", B denote the event "two heads and one tail show", C denote the event "three tails show" and D denote the event "a head shows on the first coin".
Which events are
(i) mutually exclusive
(ii) simple
(iii) compound?

12. From a group of 2 boys and 3 girls, two children are selected at random. Describe the events
(i) A: both selected children are girls.
(ii) B: the selected group consists of one boy and one girl.
(iii) C: atleast one boy is selected.
Which pair(s) of events is (are) mutually exclusive?

13. The numbers 1, 2, 3 and 4 are written separately on four slips of paper. The slips are then put in a box and mixed thoroughly. A person draws two slips from the box, one after the other, without replacement. Describe the following events:
A: The number on the first slip is larger than the number on the second slip.
B: The number on the second slip is greater than 2.
C: The sum of the numbers on the two slips is 6 or 7.
D: The number on the second slip is twice that on the first slip.
Which pair(s) of events is (are) mutually exclusive?

13.3 Mathematical Definition of Probability (Classical or A Priori Probability)

If an experiment has n exhaustive, mutually exclusive, and equally likely outcomes, then the sample space S has n sample points. If an event A contains m sample points, where 0 ≤ m ≤ n, then the probability of event A, written as P(A), is defined as:

\( P(A) = \frac{m}{n} = \frac{\text{number of outcomes favourable to A}}{\text{total number of outcomes}} \)