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Probability

Practice MCQs

Probability is an important quantitative aptitude topic that measures the chance of occurrence of an event.

Quantitative Aptitude Probability Competitive Exams

Probability is an important quantitative aptitude topic that measures the chance of occurrence of an event. It is commonly asked in competitive exams, aptitude tests, data interpretation, statistics, cards, dice, coins, balls, and selection-based problems.

What is Probability?

Probability is the measure of how likely an event is to occur. It is expressed as a number between \(0\) and \(1\), where \(0\) means impossible and \(1\) means certain.

In exams, probability questions are usually based on coins, dice, cards, balls, numbers, selections, and arrangements. The basic idea is to compare the number of favourable outcomes with the total number of possible outcomes.

Quick idea: Probability is favourable outcomes divided by total outcomes.
Term Meaning Example
Experiment An action with uncertain result Tossing a coin
Outcome Possible result of an experiment Head or Tail
Sample Space Set of all possible outcomes \(\{H,T\}\)
Event Desired outcome or group of outcomes Getting Head
Favourable Outcomes Outcomes that satisfy the condition Even number on a die: 2, 4, 6

“Probability is the ratio of favourable outcomes to total possible outcomes.”

Aptitude Tip
Key points
  • Probability always lies between \(0\) and \(1\).
  • Impossible event has probability \(0\).
  • Certain event has probability \(1\).
  • Total probability of all outcomes is \(1\).
  • Use combinations when order does not matter.
  • Use permutations when order matters.
event outcome sample space chance

Visual Understanding

These diagrams show probability scale, coin outcomes, and die outcomes.

Probability Scale
0 0.5 1 Impossible Equally likely Certain Probability always lies from 0 to 1
\[ 0 \leq P(E) \leq 1 \]

Every probability value must lie between \(0\) and \(1\).

Coin Toss
H Head T Tail Sample Space = {H, T}
\[ P(H)=\frac{1}{2} \]

A fair coin has two equally likely outcomes.

Die Outcomes
Outcomes: 1, 2, 3, 4, 5, 6
\[ P(\text{even})=\frac{3}{6}=\frac{1}{2} \]

A fair die has six equally likely outcomes.

Important Formulas and Rules

Basic Probability
\[ P(E)=\frac{\text{Favourable Outcomes}}{\text{Total Outcomes}} \]

Used when all outcomes are equally likely.

Range of Probability
\[ 0 \leq P(E) \leq 1 \]

Probability cannot be less than \(0\) or greater than \(1\).

Complement Rule
\[ P(E')=1-P(E) \]

Probability of event not happening.

Impossible Event
\[ P(E)=0 \]

Event that cannot happen.

Certain Event
\[ P(E)=1 \]

Event that is sure to happen.

Either A or B
\[ P(A\cup B)=P(A)+P(B)-P(A\cap B) \]

Used when events may overlap.

Independent Events
\[ P(A\cap B)=P(A)\times P(B) \]

Used when one event does not affect the other.

Conditional Probability
\[ P(A|B)=\frac{P(A\cap B)}{P(B)} \]

Probability of \(A\) when \(B\) has already happened.

Rule: Probability is valid only when favourable outcomes are counted from the same sample space as total outcomes.

Common Sample Spaces

Experiment Total Outcomes Sample Space / Useful Count
One coin tossed 2 \(\{H,T\}\)
Two coins tossed 4 \(\{HH,HT,TH,TT\}\)
One die thrown 6 \(\{1,2,3,4,5,6\}\)
Two dice thrown 36 \(6\times6=36\)
One card drawn from deck 52 13 cards in each suit, 4 suits
Selecting \(r\) objects from \(n\) \({}^nC_r\) Use when order does not matter
Tip: In probability, most errors happen because total outcomes are counted incorrectly.

Common Types of Probability Questions

Coin Problems

Based on heads and tails.

  • One coin
  • Two coins
  • At least one head
  • Exactly one tail
Dice Problems

Based on die outcomes.

  • Even or odd number
  • Prime number
  • Sum of two dice
  • Greater than a number
Card Problems

Based on standard deck of 52 cards.

  • Red card
  • Black card
  • Face card
  • Ace or king
Selection Problems

Based on choosing objects.

  • Balls from bag
  • Students from group
  • Committee selection
  • Use combinations
Exam approach: First write total outcomes, then count favourable outcomes. Use the same counting method for both.
Method Bank
Coin

Probability of head in one toss:

\[ P(H)=\frac{1}{2} \]
Die

Probability of even number:

\[ P(E)=\frac{3}{6}=\frac{1}{2} \]
Cards

Probability of red card:

\[ P(\text{red})=\frac{26}{52}=\frac{1}{2} \]
Complement

Probability of not getting event \(E\):

\[ P(E')=1-P(E) \]

Tip: For “at least one” questions, complement method is often faster.

Standard Deck of Cards
♥ 13 ♦ 13 ♣ 13 ♠ 13 Total cards = 52
A standard deck has 52 cards: 26 red cards, 26 black cards, 4 aces, and 12 face cards.

Step-by-Step Solving Method

Step Action Example
Step 1 Identify the experiment. Throwing one die.
Step 2 Write total outcomes. \(1,2,3,4,5,6\), total \(=6\).
Step 3 Count favourable outcomes. Even numbers: \(2,4,6\), favourable \(=3\).
Step 4 Apply probability formula. \(P(E)=\frac{3}{6}\).
Step 5 Simplify the answer. \(P(E)=\frac{1}{2}\).
Important: Always ensure that all outcomes are equally likely before using the basic probability formula.

Solved Examples

Question Method Answer
A coin is tossed once. Find the probability of getting a head. Total outcomes: \(H,T\). Favourable outcome: \(H\).
\[ P(H)=\frac{1}{2} \]
\(\frac{1}{2}\)
A die is thrown once. Find the probability of getting an even number. Even outcomes: \(2,4,6\). Total outcomes: \(6\).
\[ P(\text{even})=\frac{3}{6}=\frac{1}{2} \]
\(\frac{1}{2}\)
A die is thrown once. Find the probability of getting a prime number. Prime outcomes on die: \(2,3,5\). Total outcomes: \(6\).
\[ P(\text{prime})=\frac{3}{6}=\frac{1}{2} \]
\(\frac{1}{2}\)
A card is drawn from a standard deck. Find the probability of getting an ace. Total cards \(=52\), aces \(=4\).
\[ P(\text{ace})=\frac{4}{52}=\frac{1}{13} \]
\(\frac{1}{13}\)
A card is drawn from a standard deck. Find the probability of getting a red card. Red cards \(=26\), total cards \(=52\).
\[ P(\text{red})=\frac{26}{52}=\frac{1}{2} \]
\(\frac{1}{2}\)
Two coins are tossed. Find the probability of getting exactly one head. Sample space: \(\{HH,HT,TH,TT\}\). Exactly one head: \(HT,TH\).
\[ P(\text{exactly one head})=\frac{2}{4}=\frac{1}{2} \]
\(\frac{1}{2}\)
A bag contains 3 red balls and 5 blue balls. One ball is drawn. Find the probability of drawing a red ball. Total balls \(=3+5=8\), red balls \(=3\).
\[ P(\text{red})=\frac{3}{8} \]
\(\frac{3}{8}\)
A number is chosen from 1 to 10. Find the probability that it is divisible by 3. Numbers divisible by 3: \(3,6,9\). Total numbers \(=10\).
\[ P=\frac{3}{10} \]
\(\frac{3}{10}\)

Note: Count favourable outcomes carefully and compare them with total outcomes from the same sample space.

Common Traps and Shortcuts

Common Traps
  • Counting favourable outcomes incorrectly.
  • Using wrong total outcomes.
  • Forgetting that two dice have \(36\) outcomes, not \(12\).
  • Confusing “at least one” with “exactly one”.
  • Using permutation when selection is needed.
  • Forgetting replacement or non-replacement conditions.
Useful Shortcuts
  • One coin has \(2\) outcomes.
  • Two coins have \(4\) outcomes.
  • One die has \(6\) outcomes.
  • Two dice have \(36\) outcomes.
  • One deck has \(52\) cards.
  • For “not” questions, use complement rule.
Exam approach: Write the sample space or count total outcomes first. Then count only the favourable outcomes.

Practice

A) Multiple Choice Questions
  1. A coin is tossed once. Probability of getting tail is:
    \(\frac{1}{4}\) \(\frac{1}{3}\) \(\frac{1}{2}\) 1
  2. A die is thrown once. Probability of getting a number greater than 4 is:
    \(\frac{1}{6}\) \(\frac{1}{3}\) \(\frac{1}{2}\) \(\frac{2}{3}\)
  3. A card is drawn from a deck. Probability of getting a king is:
    \(\frac{1}{13}\) \(\frac{1}{26}\) \(\frac{1}{4}\) \(\frac{1}{2}\)
  4. A bag has 4 red and 6 blue balls. Probability of drawing a blue ball is:
    \(\frac{2}{5}\) \(\frac{3}{5}\) \(\frac{1}{5}\) \(\frac{4}{5}\)
  5. Probability of an impossible event is:
    0 \(\frac{1}{2}\) 1 2
B) Solve the Higher-Order Problems
  1. Two coins are tossed. Find the probability of getting at least one head. (Hint: Sample space has 4 outcomes.)
  2. A die is thrown. Find the probability of getting a prime number. (Hint: Prime outcomes are 2, 3, and 5.)
  3. One card is drawn from a standard deck. Find the probability of getting a face card. (Hint: Face cards are J, Q, K.)
  4. A bag contains 5 white balls and 7 black balls. One ball is drawn. Find the probability of not getting a white ball. (Hint: Use complement or count black balls.)
  5. A number is chosen from 1 to 20. Find the probability that it is a multiple of 4. (Hint: Multiples are 4, 8, 12, 16, 20.)
C) Match the Concept with the Correct Meaning
Concept Correct Meaning
Probability Measure of chance of an event
Sample Space Set of all possible outcomes
Event Desired outcome or group of outcomes
Impossible Event Event with probability 0
Certain Event Event with probability 1
Complement Event not happening
Aptitude Reminder

Probability is based on counting favourable outcomes and total outcomes. Always define the sample space clearly. In coin, dice, card, and ball problems, carefully check whether replacement is involved and whether order matters.

Task: Create five probability questions using coins, dice, cards, balls, and number selection.

Show Suggested Answers
Multiple Choice
  1. \(\frac{1}{2}\)
    A coin has two outcomes: \(H,T\). Tail is one favourable outcome.
    \[ P(T)=\frac{1}{2} \]
  2. \(\frac{1}{3}\)
    Numbers greater than \(4\) are \(5,6\). Total outcomes \(=6\).
    \[ P=\frac{2}{6}=\frac{1}{3} \]
  3. \(\frac{1}{13}\)
    There are \(4\) kings in \(52\) cards.
    \[ P(\text{king})=\frac{4}{52}=\frac{1}{13} \]
  4. \(\frac{3}{5}\)
    Blue balls \(=6\), total balls \(=4+6=10\).
    \[ P(\text{blue})=\frac{6}{10}=\frac{3}{5} \]
  5. 0
    Impossible event has probability \(0\).
Higher-Order Problems
  1. Two coins sample space: \(\{HH,HT,TH,TT\}\). At least one head: \(HH,HT,TH\).
    \[ P(\text{at least one head})=\frac{3}{4} \]
    Answer = \(\frac{3}{4}\).
  2. Prime outcomes on a die are \(2,3,5\). Total outcomes \(=6\).
    \[ P(\text{prime})=\frac{3}{6}=\frac{1}{2} \]
    Answer = \(\frac{1}{2}\).
  3. Face cards are \(J,Q,K\) in each suit. Total face cards \(=12\).
    \[ P(\text{face card})=\frac{12}{52}=\frac{3}{13} \]
    Answer = \(\frac{3}{13}\).
  4. White balls \(=5\), black balls \(=7\), total \(=12\). Not white means black.
    \[ P(\text{not white})=\frac{7}{12} \]
    Answer = \(\frac{7}{12}\).
  5. Multiples of \(4\) from \(1\) to \(20\): \(4,8,12,16,20\). Favourable \(=5\), total \(=20\).
    \[ P=\frac{5}{20}=\frac{1}{4} \]
    Answer = \(\frac{1}{4}\).
Concept Matching
  1. Probability → Measure of chance of an event
  2. Sample Space → Set of all possible outcomes
  3. Event → Desired outcome or group of outcomes
  4. Impossible Event → Event with probability 0
  5. Certain Event → Event with probability 1
  6. Complement → Event not happening
Clue Explanation

Probability questions mainly depend on correct counting. Always count favourable outcomes and total outcomes from the same experiment.

Exam tips
  • Write total outcomes first.
  • Then count favourable outcomes.
  • For “not” questions, use complement rule.
  • For selection problems, use combinations.
  • For two dice, total outcomes are \(36\).
  • Always simplify the final fraction.
Practice MCQs