Introduction
Probability measures the chance of an event occurring. In this chapter, you will learn the classical (theoretical) approach to probability. You will calculate the probability of various events related to coins, dice, cards, and real-life situations. The probability of an event always lies between 0 (impossible) and 1 (certain).
Theoretical Probability
The theoretical probability of an event E is defined as P(E) = Number of outcomes favourable to E / Total number of equally likely outcomes. This assumes all outcomes are equally likely. For example, when tossing a fair coin, the probability of heads is 1/2 because there is 1 favourable outcome (head) out of 2 equally likely outcomes (head, tail).
Key Points
- •P(E) = Favourable outcomes / Total outcomes
- •0 <= P(E) <= 1 for any event E
- •P(sure event) = 1, P(impossible event) = 0
- •P(E) + P(not E) = 1, so P(not E) = 1 - P(E)
- •All outcomes must be equally likely for this formula to apply
Worked Example
A die is thrown once. Find P(getting a number greater than 4). Total outcomes = {1, 2, 3, 4, 5, 6} = 6 Favourable outcomes = {5, 6} = 2 P(number > 4) = 2/6 = 1/3
Watch Out
When finding the probability of 'not E', use P(not E) = 1 - P(E). It is often easier to calculate the complement.
Probability with Coins and Dice
Coins and dice are the most common tools in probability problems. A single coin has 2 outcomes (H, T). Two coins have 4 outcomes (HH, HT, TH, TT). A single die has 6 outcomes (1 through 6). Two dice have 36 outcomes. For cards, a standard deck has 52 cards: 4 suits (hearts, diamonds, clubs, spades) with 13 cards each.
Key Points
- •One coin: 2 outcomes; Two coins: 4 outcomes; Three coins: 8 outcomes
- •One die: 6 outcomes; Two dice: 36 outcomes
- •A deck of cards: 52 cards, 4 suits, 13 ranks per suit
- •Red cards: 26 (hearts + diamonds), Black cards: 26 (clubs + spades)
- •Face cards: 12 (4 Jacks + 4 Queens + 4 Kings)
Worked Example
Two dice are thrown simultaneously. Find P(sum = 7). Total outcomes = 6 x 6 = 36 Favourable outcomes for sum 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) = 6 P(sum = 7) = 6/36 = 1/6
Probability with Cards
A standard deck of 52 playing cards is divided into 4 suits: hearts (red), diamonds (red), clubs (black), and spades (black). Each suit has 13 cards: Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King. Face cards are Jack, Queen, and King (total 12 face cards). Many board exam questions are based on drawing a card from a well-shuffled deck.
Key Points
- •Total cards = 52 (26 red + 26 black)
- •Each suit has 13 cards: A, 2-10, J, Q, K
- •Face cards = J + Q + K = 12 total (3 per suit)
- •Number cards = 2 through 10 = 36 total (9 per suit)
- •Aces are NOT face cards
Worked Example
A card is drawn from a deck. Find P(getting a red face card). Red face cards: Hearts (J, Q, K) + Diamonds (J, Q, K) = 6 P(red face card) = 6/52 = 3/26
Watch Out
Remember: Aces are NOT face cards. This is a common trick question in board exams.
Complementary Events and Applications
Two events are complementary if they are mutually exclusive and exhaustive — one of them must occur. P(E) + P(not E) = 1. This is extremely useful when it is easier to find the probability of the complement. For example, the probability of getting at least one head when two coins are tossed is easier to find as 1 - P(no heads).
Key Points
- •P(at least one) = 1 - P(none)
- •P(E) + P(not E) = 1
- •Use complement when 'at least one' appears in the question
- •Complementary events cover all possibilities
- •Sum of probabilities of all elementary events = 1
Worked Example
Two coins are tossed. Find P(at least one head). P(no head) = P(TT) = 1/4 P(at least one head) = 1 - P(no head) = 1 - 1/4 = 3/4
Quick Summary
- ✓P(E) = Favourable outcomes / Total outcomes
- ✓0 <= P(E) <= 1
- ✓P(E) + P(not E) = 1
- ✓One coin: 2 outcomes, one die: 6 outcomes
- ✓Standard deck: 52 cards, 4 suits, 13 per suit, 12 face cards
- ✓P(at least one) = 1 - P(none)
- ✓All outcomes must be equally likely
Key Formulas
P(E) = Number of favourable outcomes / Total number of outcomes
P(not E) = 1 - P(E)
P(sure event) = 1, P(impossible event) = 0
Coin: 2 outcomes, Die: 6 outcomes, Cards: 52
Two coins: 4 outcomes, Two dice: 36 outcomes
Face cards: 12 (J, Q, K x 4 suits), Aces are NOT face cards
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