Probability Fundamentals

Probability quantifies uncertainty and helps us make informed decisions!

WHAT IS PROBABILITY? Probability is a number between 0 and 1 that tells you how likely an event is to occur.

Probability Scale: 0 = Impossible (will never happen) 0.5 = Equally likely (50-50 chance) 1 = Certain (will always happen)

THE BASIC FORMULA:

P(event) = Number of favorable outcomes / Total number of possible outcomes

Example: Rolling a 4 on a standard die - Favorable outcomes: 1 (just the face with 4) - Total outcomes: 6 (faces numbered 1-6) - P(4) = 1/6

KEY VOCABULARY:

Outcome: A possible result of an experiment Example: Flipping a coin → outcomes: heads or tails

Sample Space: ALL possible outcomes Example: Rolling a die → sample space: {1, 2, 3, 4, 5, 6}

Event: One or more outcomes we're interested in Example: Rolling an even number → event: {2, 4, 6}

Favorable Outcomes: Outcomes that match our event Example: For "rolling even," favorable outcomes are 2, 4, 6 (3 outcomes)

CALCULATING PROBABILITY:

Example 1: Spinner An 8-section spinner has colors: Red, Red, Blue, Blue, Blue, Green, Yellow, Yellow

P(Blue) = ? - Favorable: 3 blue sections - Total: 8 sections - P(Blue) = 3/8

Example 2: Deck of Cards Standard deck has 52 cards: 13 hearts, 13 diamonds, 13 clubs, 13 spades

P(Heart) = ? - Favorable: 13 hearts - Total: 52 cards - P(Heart) = 13/52 = 1/4

Example 3: Jar of Marbles A jar has 5 red, 3 blue, 2 green marbles

P(Red) = ? - Favorable: 5 red - Total: 5 + 3 + 2 = 10 - P(Red) = 5/10 = 1/2

EXPRESSING PROBABILITY:

Probabilities can be written as: - Fractions: 1/4 - Decimals: 0.25 - Percents: 25%

All three mean the same thing!

PROBABILITY OF COMPLEMENTARY EVENTS:

Complement = the event NOT happening

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

Example: If P(Rain) = 0.3 Then P(No rain) = 1 - 0.3 = 0.7

Example: P(not rolling a 6) = 1 - 1/6 = 5/6

SPECIAL PROBABILITIES:

Impossible Event: P = 0 Example: Rolling a 7 on a standard die → P = 0/6 = 0

Certain Event: P = 1 Example: Rolling a number less than 7 on a standard die → P = 6/6 = 1

EQUALLY LIKELY OUTCOMES: When all outcomes have the same probability - Fair coin: P(Heads) = P(Tails) = 1/2 - Fair die: Each number has P = 1/6 - Random card: Each card has P = 1/52

THEORETICAL vs EXPERIMENTAL PROBABILITY:

Theoretical: Based on logic and math P(Heads) = 1/2 because there are 2 equally likely outcomes

Experimental: Based on actual trials Flip a coin 100 times, get 47 heads → P(Heads) ≈ 47/100 = 0.47

As trials increase, experimental approaches theoretical!

REAL-WORLD APPLICATIONS: - Weather forecasting (60% chance of rain) - Games of chance (lottery odds) - Medical testing (probability of disease) - Sports predictions (winning probability) - Quality control (defect rates) - Insurance (risk assessment)

Calculate probabilities and understand chance!

BASIC PROBABILITY: 1. A bag has 12 marbles: 5 red, 4 blue, 3 green. What is P(red)? 2. A spinner has 10 equal sections: 4 yellow, 3 red, 2 blue, 1 green. What is P(yellow)? 3. A standard die is rolled. What is P(rolling a 3)? 4. What is P(rolling an even number on a standard die)?

CARD DECK PROBLEMS: A standard deck has 52 cards (13 each of hearts, diamonds, clubs, spades).

5. What is P(drawing a spade)? 6. What is P(drawing a red card)? (Hearts and diamonds are red) 7. What is P(drawing a king)? (4 kings in the deck) 8. What is P(drawing the ace of hearts)?

COMPLEMENTARY EVENTS: 9. If P(snow) = 0.15, what is P(no snow)? 10. If P(winning) = 2/5, what is P(losing)? 11. On a spinner, P(red) = 1/4. What is P(not red)? 12. If P(passing) = 0.85, what is P(not passing)?

WORD PROBLEMS: 13. A jar contains 8 chocolate chip cookies, 5 oatmeal cookies, and 7 sugar cookies. If you randomly select one cookie, what is the probability it's chocolate chip?

14. In a class of 30 students, 18 play sports. If a student is randomly selected, what is the probability they play sports?

15. A bag contains tiles with letters: A, A, A, B, B, C, D, D, E. What is the probability of drawing an A?

PROBABILITY SCALE: Classify these probabilities as impossible, unlikely, equally likely, likely, or certain.

16. P = 0.01 17. P = 0.5 18. P = 0.99 19. P = 0 20. P = 1

MULTIPLE REPRESENTATIONS: 21. Express P = 3/5 as a decimal and percent. 22. Express P = 0.4 as a fraction and percent. 23. If 20% of items are defective, express this probability as a decimal and fraction.

CHALLENGE: 24. A number cube has faces labeled 1, 1, 2, 3, 3, 3. What is P(rolling a 3)?

25. Two events A and B are complementary. If P(A) = 3/8, what is P(B)?

CRITICAL THINKING: 26. You flip a fair coin 10 times and get 10 heads. What is the probability of getting heads on the next flip? Explain your reasoning.

27. A bag has 10 red marbles and 5 blue marbles. You draw a red marble and don't replace it. Has P(red) increased, decreased, or stayed the same for the next draw? Explain.