Probability

Basic probability, conditional events

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Probability is a branch of mathematics that deals with the likelihood of events occurring. It quantifies uncertainty by assigning a numerical value between 0 and 1 to each possible outcome, where 0 means impossible and 1 means certain. Probability is essential in competitive exams for solving problems involving cards, dice, coins, and real-world scenarios.

Key formulas and rules

P(E) = Number of favourable outcomes / Total number of possible outcomes

P(E) + P(not E) = 1 (Complementary events)

P(A union B) = P(A) + P(B) - P(A intersect B) (Addition rule)

P(A intersect B) = P(A) x P(B|A) = P(B) x P(A|B) (Multiplication rule)

P(A|B) = P(A intersect B) / P(B) (Conditional probability)

For independent events: P(A intersect B) = P(A) x P(B)

For mutually exclusive events: P(A union B) = P(A) + P(B)

Odds in favour = Number of favourable outcomes : Number of unfavourable outcomes

Odds against = Number of unfavourable outcomes : Number of favourable outcomes

Key concepts

Classical (Theoretical) Probability

This is the basic definition where probability is calculated as the ratio of favourable outcomes to total possible outcomes, assuming all outcomes are equally likely. For example, P(getting a head) = 1/2 when tossing a fair coin. This approach works well for simple, well-defined experiments like coin tosses, dice rolls, and drawing cards from a well-shuffled deck.

Complementary Events

The complement of an event E (denoted as E' or not E) consists of all outcomes where E does not occur. Since either E happens or it doesn't, P(E) + P(E') = 1. This is useful when calculating P(E) directly is difficult but calculating P(E') is easier. For example, P(at least one head in 3 tosses) = 1 - P(no heads) = 1 - 1/8 = 7/8.

Worked examples

Example 1

A card is drawn from a well-shuffled pack of 52 cards. What is the probability of drawing a king or a queen?

Solution

Step 1: Identify total outcomes.
Total cards = 52.

Step 2: Identify favourable outcomes.
Number of kings = 4 (one in each suit).
Number of queens = 4 (one in each suit).
Since king and queen are mutually exclusive (a card cannot be both), we add them.
Favourable outcomes = 4 + 4 = 8.

Step 3: Calculate probability.
P(king or queen) = 8/52 = 2/13.

Answer: 2/13

Example 2

Two dice are thrown simultaneously. What is the probability of getting a sum of 7?

Solution

Step 1: Calculate total possible outcomes.
Each die has 6 faces.
Total outcomes = 6 x 6 = 36.

Step 2: Find favourable outcomes (pairs that sum to 7).
(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)
Favourable outcomes = 6.

Step 3: Calculate probability.
P(sum = 7) = 6/36 = 1/6.

Answer: 1/6

Example 3

A bag contains 4 red, 5 blue, and 3 green balls. Two balls are drawn at random without replacement. What is the probability that both are red?

Solution

Step 1: Calculate total balls and probability of first red.
Total balls = 4 + 5 + 3 = 12.
P(first is red) = 4/12 = 1/3.

Step 2: Calculate conditional probability of second red.
After drawing one red ball, 3 red balls remain out of 11 total balls.
P(second is red | first was red) = 3/11.

Step 3: Apply multiplication rule.
P(both red) = P(first red) x P(second red | first red)
P(both red) = (4/12) x (3/11) = 12/132 = 1/11.

Answer: 1/11

Example 4

Three coins are tossed simultaneously. What is the probability of getting at least 2 heads?

Solution

Step 1: Calculate total outcomes.
Total outcomes = 2^3 = 8 (HHH, HHT, HTH, THH, HTT, THT, TTH, TTT).

Step 2: Identify favourable outcomes for 'at least 2 heads'.
This means 2 heads or 3 heads.
2 heads: HHT, HTH, THH (3 outcomes).
3 heads: HHH (1 outcome).
Total favourable = 3 + 1 = 4.

Step 3: Calculate probability.
P(at least 2 heads) = 4/8 = 1/2.

Alternative method using complementary events:
P(at least 2 heads) = 1 - P(0 heads) - P(1 head)
P(0 heads) = 1/8, P(1 head) = 3/8
P(at least 2 heads) = 1 - 1/8 - 3/8 = 4/8 = 1/2.

Answer: 1/2

Probability

Key formulas and rules

Key concepts

Classical (Theoretical) Probability

Complementary Events

Mutually Exclusive Events

Independent Events

Conditional Probability

Cards, Dice, and Coins

Worked examples

Example 1

Example 2

Example 3

Example 4

Representative solved questions

Question 1

Question 2

Question 3

Common mistakes and useful habits

Ready to test your understanding?

Key formulas and rules

Key concepts

Classical (Theoretical) Probability

Complementary Events

Mutually Exclusive Events

Independent Events

Conditional Probability

Cards, Dice, and Coins

Worked examples

Example 1

Example 2

Example 3

Example 4

Representative solved questions

Question 1

Question 2

Question 3

Common mistakes and useful habits

Ready to test your understanding?

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