Probability Basics

What is Probability?

Probability is the measure of how likely an event is to occur.

Scale: 0 to 1 (or 0% to 100%)

• 0 (or 0%): Impossible event
• 1 (or 100%): Certain event
• 0.5 (or 50%): Equally likely to happen or not

Real-life examples:

• Probability of rain: 70%
• Probability of heads on coin flip: 50%
• Probability of rolling a 6: 1/6 ≈ 16.67%

Basic Probability Formula

Formula: P(event) = (number of favorable outcomes) / (total number of possible outcomes)

Example 1: Rolling a standard die What's the probability of rolling a 4?

Favorable outcomes: 1 (only the 4) Total outcomes: 6 (numbers 1-6) P(rolling 4) = 1/6 ≈ 0.167 or 16.67%

Example 2: Drawing a card from standard deck (52 cards) What's the probability of drawing a heart?

Favorable: 13 hearts Total: 52 cards P(heart) = 13/52 = 1/4 = 0.25 or 25%

Expressing Probability

Probability can be written as:

• Fraction: 1/4
• Decimal: 0.25
• Percent: 25%
• Ratio: 1:3 (1 success to 3 failures)

All express the same likelihood!

Sample Space

The sample space is the set of ALL possible outcomes.

Example 1: Flipping a coin Sample space: {Heads, Tails} Total outcomes: 2

Example 2: Rolling a die Sample space: {1, 2, 3, 4, 5, 6} Total outcomes: 6

Example 3: Spinning spinner with colors Sample space: {Red, Blue, Green, Yellow} Total outcomes: 4

Simple Events

A simple event is a single outcome.

Example: Rolling a die

• Rolling a 3 is a simple event
• P(3) = 1/6

Compound Events

A compound event combines two or more simple events.

Example: Rolling a die

• Rolling an even number: {2, 4, 6}
• P(even) = 3/6 = 1/2

Example 2: Drawing a card

• Drawing a face card (J, Q, K)
• Favorable: 12 cards (4 Jacks, 4 Queens, 4 Kings)
• P(face card) = 12/52 = 3/13

Theoretical vs. Experimental Probability

Theoretical probability: Based on reasoning and math

• Formula: favorable/total
• Example: P(heads) = 1/2

Experimental probability: Based on actual experiments

• Formula: successes/trials
• Example: Flipped coin 100 times, got 48 heads
  • P(heads) = 48/100 = 0.48

Law of Large Numbers: As trials increase, experimental approaches theoretical.

Complementary Events

Complement of event A: all outcomes that are NOT A Notation: A' or Ā (read as "not A")

Formula: P(A') = 1 - P(A)

Example 1: Rolling a die P(rolling 5) = 1/6 P(NOT rolling 5) = 1 - 1/6 = 5/6

Example 2: Drawing a card P(heart) = 1/4 P(not heart) = 1 - 1/4 = 3/4

Why useful? Sometimes easier to find P(not happening)!

Probability of Impossible and Certain Events

Impossible event: P = 0

• Example: Rolling a 7 on standard die
• P(rolling 7) = 0/6 = 0

Certain event: P = 1

• Example: Rolling a number 1-6 on standard die
• P(1-6) = 6/6 = 1

Between: All other probabilities

• 0 < P < 1

"AND" vs. "OR" Probability

"AND" (both events happen):

• More restrictive
• Usually smaller probability
• Example: Drawing Ace AND Heart = 1/52

"OR" (at least one event happens):

• Less restrictive
• Usually larger probability
• Example: Drawing Ace OR Heart = 16/52

(We'll explore these more in compound probability!)

Equally Likely Outcomes

Basic probability formula assumes equally likely outcomes.

Example 1: Fair coin Each outcome (H or T) equally likely P(H) = P(T) = 1/2

Example 2: Biased coin (70% heads) Outcomes NOT equally likely P(H) = 0.7, P(T) = 0.3

When to use basic formula:

• Fair dice, coins
• Random selection
• Each outcome has same chance

Probability with Spinners

Example: Spinner divided into 8 equal sections: 3 red, 2 blue, 2 green, 1 yellow

P(red) = 3/8 P(blue) = 2/8 = 1/4 P(green) = 2/8 = 1/4 P(yellow) = 1/8

Check: 3/8 + 2/8 + 2/8 + 1/8 = 8/8 = 1 ✓

Probability with Marbles

Example: Bag contains 5 red, 3 blue, 2 green marbles

Total: 10 marbles

P(red) = 5/10 = 1/2 P(blue) = 3/10 P(green) = 2/10 = 1/5 P(not green) = 1 - 1/5 = 4/5

Multiple Trials

Example: Roll die twice How many total outcomes?

First roll: 6 outcomes Second roll: 6 outcomes Total: 6 × 6 = 36 outcomes

Sample space size: 36 different pairs (1,1), (1,2), ..., (6,6)

Tree Diagrams

Tree diagram shows all possible outcomes visually.

Example: Flip coin twice

Possible paths:

• First flip H, Second flip H → Outcome: HH
• First flip H, Second flip T → Outcome: HT
• First flip T, Second flip H → Outcome: TH
• First flip T, Second flip T → Outcome: TT

Sample space: {HH, HT, TH, TT} Total outcomes: 4 P(both heads) = 1/4 P(exactly one head) = 2/4 = 1/2

Counting Principle

Multiplication Principle: If one event has m outcomes and another has n outcomes, together they have m × n outcomes.

Example: Choose outfit

• 3 shirts
• 4 pants Total outfits: 3 × 4 = 12

Example 2: Create PIN

• 4 digits
• Each digit: 0-9 (10 choices) Total PINs: 10 × 10 × 10 × 10 = 10,000

Odds

Odds compare favorable to unfavorable outcomes.

Odds in favor: favorable : unfavorable

Example: Rolling 5 on die Favorable: 1 Unfavorable: 5 Odds in favor: 1:5

Odds against: unfavorable : favorable Odds against: 5:1

Converting between probability and odds: If P(A) = 1/6, then odds = 1:5

Independent vs. Dependent Events

Independent: One event doesn't affect the other

• Example: Flip coin twice
• First flip doesn't change second flip

Dependent: One event affects the other

• Example: Draw two cards without replacement
• First card removed affects second draw

(We'll explore more in compound probability!)

Real-World Applications

Weather: 60% chance of rain

• P(rain) = 0.6
• P(no rain) = 0.4

Medicine: 95% effective treatment

• P(success) = 0.95
• P(failure) = 0.05

Games: Lottery odds

• Mega Millions: about 1 in 302 million
• P(winning) ≈ 0.0000000033

Quality control: 2% defect rate

• P(defective) = 0.02
• P(not defective) = 0.98

Probability in Games

Example 1: Standard deck of cards

• P(Ace) = 4/52 = 1/13
• P(Spade) = 13/52 = 1/4
• P(Red card) = 26/52 = 1/2

Example 2: Roulette (American)

• 38 slots (1-36, 0, 00)
• 18 red, 18 black, 2 green
• P(red) = 18/38 ≈ 0.474

Simulations

Simulation: Using random process to model probability.

Example: Simulate coin flips

• Use random number generator
• 0-4 = Heads, 5-9 = Tails
• Run 100 trials
• Count heads: experimental probability

Why simulate?

• Test theoretical predictions
• Model complex situations
• Cheaper than real experiments

Common Misconceptions

Gambler's Fallacy: "Coin landed heads 5 times, so tails is due!" → FALSE: Each flip is independent, still 50/50

Law of Averages: "If I keep playing, I'll eventually win!" → FALSE: Past outcomes don't guarantee future results

Hot Hand: "I'm on a winning streak, so I'll keep winning!" → FALSE: In random events, past luck doesn't predict future

Probability Rules

Rule 1: 0 ≤ P(A) ≤ 1 for any event A

Rule 2: P(certain event) = 1

Rule 3: P(impossible event) = 0

Rule 4: P(A) + P(A') = 1

Rule 5: Sum of all probabilities in sample space = 1

Making Predictions

Expected frequency = Probability × Number of trials

Example: Roll die 60 times. How many 4s expected? P(4) = 1/6 Expected: (1/6) × 60 = 10 times

Note: This is EXPECTED, not guaranteed! Actual results will vary.

Probability in Surveys

Example: Survey of 500 students

• 200 prefer pizza
• 150 prefer burgers
• 150 prefer tacos

If one student chosen randomly: P(pizza) = 200/500 = 2/5 = 0.4 P(burger) = 150/500 = 3/10 = 0.3 P(taco) = 150/500 = 3/10 = 0.3

Geometric Probability

Geometric probability: Based on area, length, or volume.

Example: Dartboard

• Circle radius 10 (area = 100π)
• Bullseye radius 2 (area = 4π)
• P(bullseye) = 4π/100π = 4/100 = 1/25

Example 2: Number line 0-10 Target region: 3-5 (length = 2) P(landing in target) = 2/10 = 1/5

Common Mistakes to Avoid

1. Adding probabilities incorrectly P(A or B) ≠ always P(A) + P(B) when events overlap

2. Assuming independence Drawing cards without replacement = dependent

3. Confusing probability with certainty 70% chance ≠ "will definitely happen"

4. Forgetting to simplify fractions Express 5/10 as 1/2

5. Using wrong denominator Make sure you count ALL possible outcomes

6. Negative probability Probability can never be negative!

Quick Reference

Basic Formula: P(event) = favorable/total

Complement: P(not A) = 1 - P(A)

Range: 0 ≤ P ≤ 1

Certain event: P = 1 Impossible event: P = 0

Expected frequency: Probability × trials

Practice Strategy

Level 1: Single events (coin, die, card) Level 2: Compound events (even number, face card) Level 3: Complements (not rolling 5) Level 4: Multiple trials (two coins, two dice) Level 5: Real-world applications

Tips for Success

• List all possible outcomes first
• Make sure outcomes are equally likely
• Simplify fractions
• Check that probability is between 0 and 1
• Use complement when easier
• Draw tree diagrams for multiple events
• Practice with dice, cards, and coins
• Understand theoretical vs. experimental
• Don't fall for gambler's fallacy
• Remember: probability predicts long-term trends, not individual events