Probability
Practice MCQsProbability is the branch of mathematics that deals with uncertainty and chance. It helps measure how likely an event is to happen.
Probability is the branch of mathematics that deals with uncertainty and chance. It helps measure how likely an event is to happen. This chapter covers random experiments, outcomes, sample space, events, mutually exclusive and exhaustive events, impossible and certain events, union and intersection of events, complementary events, classical and statistical probability, elementary theorems, conditional probability, Bayes' theorem, random variables and binomial distribution.
What is Probability?
Probability is a numerical measure of the chance of occurrence of an event. It lies between 0 and 1. A probability of 0 means the event is impossible, while a probability of 1 means the event is certain.
In competitive exams, probability questions are often based on coins, dice, cards, balls in a bag, selection problems, conditional probability, and simple binomial distribution models.
| Concept | Meaning | Example |
|---|---|---|
| Random Experiment | An experiment whose result cannot be predicted with certainty. | Tossing a coin, rolling a die. |
| Outcome | A possible result of a random experiment. | Head in coin toss, 4 in die roll. |
| Sample Space | Set of all possible outcomes. | For a die: \(S=\{1,2,3,4,5,6\}\) |
| Event | A subset of the sample space. | Getting an even number: \(\{2,4,6\}\) |
| Probability | Measure of chance of an event. | \(P(E)=\frac{\text{favourable outcomes}}{\text{total outcomes}}\) |
“Probability converts uncertainty into a measurable number.”
Key points
- Probability always lies between 0 and 1.
- Sample space contains all possible outcomes.
- An event is a subset of the sample space.
- Mutually exclusive events cannot occur together.
- Exhaustive events cover the whole sample space.
- Complementary events have probabilities adding to 1.
- Conditional probability uses given information.
- Bayes' theorem reverses conditional probability.
- Binomial distribution applies to repeated independent trials.
Core Formula Bank
Tip: Most basic probability questions first require identifying sample space and favourable outcomes.
Probability Method Selection Guide
Probability questions become easier when the student first identifies the experiment, sample space, event type and keyword used in the question. The table below helps choose the correct rule quickly.
| Question Type | What to Use | Typical Clue |
|---|---|---|
| Simple probability | \(P(E)=\frac{n(E)}{n(S)}\) | Coin, die, card, ball, direct favourable outcomes |
| Sample space question | List all possible outcomes | Asked to write or count all possible results |
| Impossible event | \(P(E)=0\) | Event cannot occur, such as getting 7 on a die |
| Certain event | \(P(E)=1\) | Event must occur, such as getting a number less than 7 on a die |
| Complement question | \(P(E')=1-P(E)\) | Words like “not”, “at least one”, “none”, “not selected” |
| Union question | \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\) | Words like “A or B” |
| Mutually exclusive events | \(P(A\cup B)=P(A)+P(B)\) | Events cannot occur together |
| Intersection question | Common outcomes or multiplication rule | Words like “A and B” |
| Independent events | \(P(A\cap B)=P(A)P(B)\) | One event does not affect the other |
| Conditional probability | \(P(A|B)=\frac{P(A\cap B)}{P(B)}\) | Words like “given that”, “if B has occurred” |
| Bayes' theorem | Reverse conditional probability | Cause is asked after an effect is observed |
| Random variable question | Assign numerical values to outcomes | Number of heads, number obtained, success/failure value |
| Binomial distribution | \(P(X=r)={}^{n}C_r p^r q^{n-r}\) | Repeated independent trials with only success/failure |
Random Experiment, Outcomes and Sample Space
A random experiment is an experiment whose outcome is not known in advance, even though all possible outcomes are known. The set of all possible outcomes is called the sample space.
| Experiment | Sample Space | Number of Outcomes |
|---|---|---|
| Tossing one coin | \(S=\{H,T\}\) | 2 |
| Tossing two coins | \(S=\{HH,HT,TH,TT\}\) | 4 |
| Rolling one die | \(S=\{1,2,3,4,5,6\}\) | 6 |
| Drawing one card from a standard deck | All 52 cards | 52 |
| Selecting one ball from a bag | All balls in the bag | Total number of balls |
Events and Types of Events
An event is a subset of the sample space. Events can be classified in different ways depending on their possibility, relation and structure.
| Type of Event | Meaning | Example |
|---|---|---|
| Elementary Event | An event containing exactly one outcome. | Getting 4 on a die: \(\{4\}\) |
| Composite Event | An event containing more than one outcome. | Getting an even number: \(\{2,4,6\}\) |
| Impossible Event | An event that cannot occur. | Getting 7 on a die. |
| Certain Event | An event that must occur. | Getting a number less than 7 on a die. |
| Mutually Exclusive Events | Events that cannot occur together. | Getting even and odd number in one die throw. |
| Exhaustive Events | Events whose union gives the whole sample space. | Getting odd or even on a die. |
| Complementary Events | One event occurs when the other does not occur. | Getting head and not getting head. |
Union, Intersection and Complement of Events
Events can be combined using set operations. These operations are important for solving probability questions involving “or”, “and”, and “not”.
| Operation | Meaning | Probability Rule |
|---|---|---|
| Union \(A\cup B\) | Event A or event B or both occur. | \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\) |
| Intersection \(A\cap B\) | Both event A and event B occur. | Used in common outcomes. |
| Complement \(A'\) | Event A does not occur. | \(P(A')=1-P(A)\) |
| Mutually Exclusive Events | \(A\cap B=\varnothing\) | \(P(A\cup B)=P(A)+P(B)\) |
| Exhaustive Events | \(A\cup B\cup C\cup \cdots =S\) | Total probability is 1. |
Definition of Probability: Classical and Statistical
Probability can be defined in different ways. The two basic definitions are classical probability and statistical or empirical probability.
| Definition | Formula | Example |
|---|---|---|
| Classical Probability | \[ P(E)=\frac{\text{Number of favourable outcomes}}{\text{Total number of equally likely outcomes}} \] | Probability of getting 3 on a die: \[ P(E)=\frac{1}{6} \] |
| Statistical Probability | \[ P(E)=\frac{\text{Number of times event occurs}}{\text{Total number of trials}} \] | If head appears 48 times in 100 tosses, estimated probability of head: \[ \frac{48}{100}=0.48 \] |
Elementary Theorems on Probability
Basic theorems of probability help solve problems involving complements, union, intersection, mutually exclusive events and independent events.
| Theorem / Rule | Formula | Use |
|---|---|---|
| Range Rule | \(0\leq P(E)\leq 1\) | Probability is never negative or greater than 1. |
| Impossible Event | \(P(\varnothing)=0\) | Event cannot occur. |
| Certain Event | \(P(S)=1\) | Event must occur. |
| Complement Rule | \(P(E')=1-P(E)\) | Used for “not” questions. |
| Addition Rule | \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\) | Used for “A or B”. |
| Mutually Exclusive Addition Rule | \(P(A\cup B)=P(A)+P(B)\) | When \(A\cap B=\varnothing\). |
| Independent Events | \(P(A\cap B)=P(A)P(B)\) | One event does not affect another. |
Conditional Probability
Conditional probability is the probability of an event occurring given that another event has already occurred.
| Expression | Meaning | Read As |
|---|---|---|
| \(P(A|B)\) | Probability of A when B is already known to have occurred. | Probability of A given B. |
| \(P(B|A)\) | Probability of B when A is already known to have occurred. | Probability of B given A. |
| \(P(A\cap B)\) | Probability that both A and B occur. | Probability of A and B. |
Bayes' Theorem
Bayes' theorem is used to find the probability of a cause when an effect is known. It reverses conditional probability.
| Term | Meaning |
|---|---|
| \(P(A_i)\) | Prior probability of cause \(A_i\). |
| \(P(B|A_i)\) | Probability of observed event \(B\) when cause \(A_i\) has occurred. |
| \(P(A_i|B)\) | Revised probability of cause \(A_i\) after observing \(B\). |
Random Variable as a Function on Sample Space
A random variable is a function that assigns a numerical value to each outcome of a random experiment. It converts outcomes into numbers.
| Experiment | Sample Space | Random Variable Example |
|---|---|---|
| Tossing two coins | \(\{HH,HT,TH,TT\}\) | Let \(X\) be number of heads. Then \(X=2,1,1,0\). |
| Rolling one die | \(\{1,2,3,4,5,6\}\) | Let \(X\) be the number obtained. Then \(X=1,2,3,4,5,6\). |
| Drawing a card | 52 cards | Let \(X=1\) if card is an ace, and \(X=0\) otherwise. |
Binomial Distribution
A random variable follows a binomial distribution when an experiment consists of a fixed number of independent trials, each trial has only two outcomes, and the probability of success remains constant.
| Condition | Meaning | Example |
|---|---|---|
| Fixed Number of Trials | The experiment is repeated \(n\) times. | Tossing a coin 5 times. |
| Two Outcomes | Each trial has success or failure. | Head or tail, defective or non-defective. |
| Independent Trials | One trial does not affect another. | Repeated coin tosses. |
| Constant Probability | Probability of success remains \(p\) for every trial. | For a fair coin, \(p=\frac{1}{2}\). |
Step-by-Step Solving Method
| Step | Action | Example Focus |
|---|---|---|
| Step 1 | Identify the random experiment. | Coin, die, cards, balls, selection. |
| Step 2 | Write the sample space or count total outcomes. | For die, total outcomes \(=6\). |
| Step 3 | Identify favourable outcomes. | Even numbers: \(\{2,4,6\}\). |
| Step 4 | Choose the correct rule. | Classical probability, addition rule, conditional probability, Bayes' theorem. |
| Step 5 | Simplify and check range. | Probability must lie between 0 and 1. |
Solved Examples
| Question | Method | Answer |
|---|---|---|
| A die is rolled once. Find the probability of getting an even number. |
Sample space: \(\{1,2,3,4,5,6\}\) Favourable outcomes: \(\{2,4,6\}\) \[ P(E)=\frac{3}{6}=\frac{1}{2} \] |
\(\frac{1}{2}\) |
| A coin is tossed twice. Find the probability of getting exactly one head. |
Sample space: \(\{HH,HT,TH,TT\}\) Favourable outcomes: \(\{HT,TH\}\) \[ P(E)=\frac{2}{4}=\frac{1}{2} \] |
\(\frac{1}{2}\) |
| A card is drawn from a deck of 52 cards. Find probability of getting an ace. | Number of aces \(=4\), total cards \(=52\). \[ P(E)=\frac{4}{52}=\frac{1}{13} \] | \(\frac{1}{13}\) |
| If \(P(A)=0.4\), find \(P(A')\). | \[ P(A')=1-P(A)=1-0.4=0.6 \] | 0.6 |
| If \(P(A)=0.5\), \(P(B)=0.4\), and \(P(A\cap B)=0.2\), find \(P(A\cup B)\). | \[ P(A\cup B)=P(A)+P(B)-P(A\cap B) \] \[ =0.5+0.4-0.2=0.7 \] | 0.7 |
| If \(P(A\cap B)=0.18\) and \(P(B)=0.6\), find \(P(A|B)\). | \[ P(A|B)=\frac{P(A\cap B)}{P(B)} = \frac{0.18}{0.6}=0.3 \] | 0.3 |
| A fair coin is tossed 3 times. Find probability of exactly 2 heads. | Here \(n=3\), \(r=2\), \(p=\frac{1}{2}\), \(q=\frac{1}{2}\). \[ P(X=2)={}^{3}C_2 \left(\frac{1}{2}\right)^2 \left(\frac{1}{2}\right)^1 \] \[ =3 \times \frac{1}{8}=\frac{3}{8} \] | \(\frac{3}{8}\) |
| A die is thrown 4 times. Find probability of getting exactly one six. | Here \(n=4\), \(r=1\), \(p=\frac{1}{6}\), \(q=\frac{5}{6}\). \[ P(X=1)={}^{4}C_1 \left(\frac{1}{6}\right) \left(\frac{5}{6}\right)^3 \] \[ =4 \times \frac{1}{6}\times \frac{125}{216} = \frac{500}{1296} = \frac{125}{324} \] | \(\frac{125}{324}\) |
Note: Always identify whether the question is based on simple probability, union, complement, conditional probability or binomial distribution.
Common Traps and Shortcuts
Common Traps
- Forgetting to write the correct sample space.
- Counting favourable outcomes incorrectly.
- Confusing mutually exclusive events with independent events.
- Using addition rule without subtracting intersection.
- Forgetting that complement probability is \(1-P(E)\).
- Ignoring the “given that” condition in conditional probability.
- Applying Bayes' theorem without exhaustive events.
- Using binomial distribution when trials are not independent.
- Forgetting \(q=1-p\) in binomial distribution.
Useful Shortcuts
- For one die, total outcomes are 6.
- For two coins, total outcomes are 4.
- For three coins, total outcomes are 8.
- For a card deck, total outcomes are 52.
- For “not” questions, use complement rule.
- For “A or B”, use union rule.
- For “A and B”, look for intersection.
- For repeated independent trials, check binomial conditions.
- Probability answer must always lie between 0 and 1.
Practice
A) Multiple Choice Questions
-
Probability of an impossible event is:
0 1 \(\frac{1}{2}\) Greater than 1
-
Sample space of rolling one die contains:
2 outcomes 4 outcomes 6 outcomes 52 outcomes
-
If \(P(A)=0.7\), then \(P(A')\) is:
0.7 0.3 1.7 0
-
Conditional probability \(P(A|B)\) means:
Probability of A or B Probability of A given B Probability of B given A Probability of not A
-
In binomial distribution, \(q\) is equal to:
\(p\) \(1-p\) \(n-p\) \(p^2\)
B) Solve the Higher-Order Problems
- A die is rolled once. Find the probability of getting a prime number. (Hint: Prime numbers on a die are 2, 3 and 5.)
- A coin is tossed three times. Find the probability of getting all heads. (Hint: Total outcomes \(=8\).)
- If \(P(A)=0.6\), \(P(B)=0.5\), and \(P(A\cap B)=0.2\), find \(P(A\cup B)\). (Hint: Use addition rule.)
- If \(P(A\cap B)=0.24\) and \(P(B)=0.8\), find \(P(A|B)\). (Hint: Use conditional probability.)
- A fair coin is tossed 4 times. Find the probability of exactly 3 heads. (Hint: Use binomial distribution.)
C) Match the Concept with the Correct Rule
| Concept | Correct Rule / Meaning |
|---|---|
| Sample Space | Set of all possible outcomes |
| Elementary Event | Event containing exactly one outcome |
| Impossible Event | Event with probability 0 |
| Certain Event | Event with probability 1 |
| Complement Rule | \(P(E')=1-P(E)\) |
| Addition Rule | \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\) |
| Conditional Probability | \(P(A|B)=\frac{P(A\cap B)}{P(B)}\) |
| Binomial Distribution | \(P(X=r)={}^{n}C_r p^r q^{n-r}\) |
Probability Reminder
Probability measures chance. Begin every problem by identifying the random experiment, sample space and favourable outcomes. Use complement, union, intersection, conditional probability, Bayes' theorem or binomial distribution depending on the structure of the question. Always check that the final probability lies between 0 and 1.
Task: Create five Probability questions using one question each from sample space, complement, union, conditional probability, Bayes' theorem and binomial distribution.
Show Suggested Answers
Multiple Choice
-
0
Probability of an impossible event is 0. -
6 outcomes
A die has sample space \(S=\{1,2,3,4,5,6\}\). -
0.3
\[ P(A')=1-P(A)=1-0.7=0.3 \] -
Probability of A given B
\(P(A|B)\) means probability of A when B is already known to have occurred. -
\(1-p\)
In binomial distribution, \(q=1-p\).
Higher-Order Problems
-
Prime numbers on a die are \(2,3,5\).
Favourable outcomes \(=3\), total outcomes \(=6\). \[ P=\frac{3}{6}=\frac{1}{2} \] Answer = \(\frac{1}{2}\). - Tossing three coins gives \(8\) outcomes. All heads is only \(HHH\). \[ P=\frac{1}{8} \] Answer = \(\frac{1}{8}\).
- \[ P(A\cup B)=P(A)+P(B)-P(A\cap B) \] \[ =0.6+0.5-0.2=0.9 \] Answer = 0.9.
- \[ P(A|B)=\frac{P(A\cap B)}{P(B)} = \frac{0.24}{0.8}=0.3 \] Answer = 0.3.
- Here \(n=4\), \(r=3\), \(p=\frac{1}{2}\), \(q=\frac{1}{2}\). \[ P(X=3)={}^{4}C_3 \left(\frac{1}{2}\right)^3 \left(\frac{1}{2}\right)^1 \] \[ =4 \times \frac{1}{16} = \frac{1}{4} \] Answer = \(\frac{1}{4}\).
Concept Matching
- Sample Space → Set of all possible outcomes
- Elementary Event → Event containing exactly one outcome
- Impossible Event → Event with probability 0
- Certain Event → Event with probability 1
- Complement Rule → \(P(E')=1-P(E)\)
- Addition Rule → \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\)
- Conditional Probability → \(P(A|B)=\frac{P(A\cap B)}{P(B)}\)
- Binomial Distribution → \(P(X=r)={}^{n}C_r p^r q^{n-r}\)
Clue Explanation
Probability questions are solved by identifying the experiment, sample space and event. Use simple probability for direct favourable outcomes, complement rule for “not” questions, addition rule for “or” questions, conditional probability for “given that” questions and binomial distribution for repeated independent trials with two outcomes.
Exam tips
- Write sample space wherever possible.
- Count favourable outcomes carefully.
- Use complement rule for “not” questions.
- Use addition rule for “or” questions.
- Use intersection for “and” questions.
- Read “given that” as conditional probability.
- Use Bayes' theorem when cause is asked after observing an effect.
- Use binomial distribution for repeated independent trials.
- Final probability must be between 0 and 1.