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Door 18 · Grade 8

📊 Probability

Probability measures likelihood from 0 (impossible) to 1 (certain)

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Experimental and Theoretical Probability

What is Probability?

Probability is the measure of how likely an event is to occur. It is a number between 0 and 1 (or 0% and 100%). A probability of 0 means the event is impossible, while a probability of 1 means the event is certain.

Two Types of Probability:

TypeDefinitionFormulaExample
Experimental ProbabilityBased on actual experiments or observations\(Number of times event occursTotal number of trials\)Flipping a coin 50 times and counting heads
Theoretical ProbabilityBased on mathematical reasoning without experiments\(Number of favorable outcomesTotal number of possible outcomes\)Calculating probability of rolling a 3 on a fair die

Key Vocabulary:

• Experiment: A repeatable process that gives results (e.g., rolling a die, flipping a coin)

• Outcome: A possible result of an experiment (e.g., heads, tails, rolling a 4)

• Event: A set of outcomes we are…

Worked example

Problem: A coin is flipped 200 times. Heads appears 108 times. Find the experimental probability of getting heads. Also, state the theoretical probability.

  1. Step 1: Experimental probability = \(Number of headsTotal flips\)
  2. Step 2: = \(108/200 = 54/100 = 0.54\)
  3. Step 3: Theoretical probability = \(1/2 = 0.5\) (since fair coin has 2 equally likely outcomes)

Answer: Step 3: Theoretical probability = \(1/2 = 0.5\) (since fair coin has 2 equally likely outcomes)

Simple Events and Likelihood of Events

What is a Simple Event?

A simple event is an event that has only one outcome. For example, rolling a 3 on a die is a simple event. Drawing a king of hearts from a deck is a simple event.

What is an Event?

An event can be a single outcome or a set of outcomes. Events can be categorized by their likelihood.

Types of Events by Likelihood:

LikelihoodDescriptionProbabilityExample
ImpossibleCannot happen0Rolling a 7 on a standard die
UnlikelySmall chance of happeningBetween 0 and 0.5Drawing an ace (4/52 ≈ 0.077)
Even ChanceEqually likely to happen or not0.5Getting heads on a coin flip
LikelyGood chance of happeningBetween 0.5 and 1Drawing a non-ace (48/52 ≈ 0.923)
CertainWill definitely happen1The sun will rise tomorrow

Complement of an Event:

The complement of an event A (written as \(A'\) or \(A\)) includes all outcomes NOT in A.

\(P(A) + P(A') = 1\) or \(P(A') =…

Worked example

Problem: A bag contains tickets numbered 1 to 20. A ticket is drawn at random. Find the probability that the number drawn is: (i) even (ii) odd (iii) a prime number (iv) greater than 15

  1. Step 1: Total possible outcomes = 20
  2. Step 2: (i) Even numbers: 2,4,6,8,10,12,14,16,18,20 → 10 numbers
  3. P(even) = \(10/20 = 1/2 = 0.5\)
  4. Step 3: (ii) Odd numbers: 1,3,5,7,9,11,13,15,17,19 → 10 numbers
  5. P(odd) = \(10/20 = 1/2 = 0.5\)

Answer: P(odd) = \(10/20 = 1/2 = 0.5\)

Probability Games and Applications

Probability Games

Probability games use chance to determine outcomes. Understanding probability helps players make better decisions and understand their chances of winning.

Common Probability Games:

GameProbability ConceptExample
Dice gamesRolling specific numbersCraps, Monopoly
Card gamesDrawing specific cardsPoker, Blackjack
Spinner gamesSpinning to land on sectionsBoard games, Wheel of Fortune
Lottery/raffleChance of winningRaffle tickets
Coin toss gamesFairness of tossesStarting a football match

Fair Games vs Unfair Games:

• Fair game: Each player has equal probability of winning

• Unfair game: One player has a higher probability of winning

Calculating Expected Value:

Expected value = (Probability of winning × Amount won) - (Probability of losing × Amount lost)

If expected value = 0, the game is fair.

If expected value > 0, the game favors the player.…

Worked example

Problem: In a raffle, 200 tickets are sold. You buy 5 tickets. What is the probability that you win the first prize? If the first prize is ₹1000, what is your expected winning?

  1. Step 1: P(win) = \(Your ticketsTotal tickets = 5/200 = 1/40 = 0.025\)
  2. Step 2: Expected winning = Probability × Prize amount
  3. Step 3: Expected winning = \(0.025 × 1000 = ₹25\)

Answer: Step 3: Expected winning = \(0.025 × 1000 = ₹25\)