🎯 Calculating Basic Probabilities

1. The Probability Formula

In the previous lesson, we learned what probability is. Now let's dive into how to actually calculate it! The basic probability formula is the foundation for all probability calculations.

P(Event) = Number of Favorable Outcomes / Total Number of Possible Outcomes
Key Terms:
• Favorable outcomes: The outcomes that match what we're looking for
• Total outcomes: All possible outcomes in the sample space
Example 1: Rolling a Die

What is the probability of rolling a number greater than 4 on a standard six-sided die?

Solution:

• Sample space: {1, 2, 3, 4, 5, 6} → Total outcomes = 6

• Numbers greater than 4: {5, 6} → Favorable outcomes = 2

• P(number > 4) = 2/6 = 1/3

📝 Practice Question

A bag contains 5 red balls, 3 blue balls, and 2 green balls. What is the probability of randomly selecting a red ball?
A) 1/5
B) 1/2
C) 5/8
D) 3/10

2. Fractions, Decimals, and Percentages

Probabilities can be expressed in three different ways: as fractions, decimals, or percentages. It's important to be comfortable converting between these forms!

Converting Probabilities
Fraction
3/4
Decimal
0.75
Percentage
75%
Conversion Rules:
• Fraction → Decimal: Divide the numerator by the denominator
• Decimal → Percentage: Multiply by 100 and add the % symbol
• Percentage → Decimal: Divide by 100
• Fraction → Percentage: Convert to decimal first, then to percentage
Example 2: Converting Forms

Express the probability 7/20 as a decimal and a percentage.

Solution:

• Decimal: 7 ÷ 20 = 0.35

• Percentage: 0.35 × 100 = 35%

📝 Practice Question

If the probability of rain tomorrow is 0.60, what is this probability as a percentage?
A) 6%
B) 0.6%
C) 60%
D) 600%

3. The Probability Scale (0 to 1)

All probabilities fall on a scale from 0 to 1 (or 0% to 100%). Understanding this scale helps us interpret what probability values mean in real-world terms.

0 0.25 0.5 0.75 1 Impossible Unlikely Equally Likely Likely Certain
Understanding the Scale:
• P = 0: The event is impossible (will never happen)
• 0 < P < 0.5: The event is unlikely but possible
• P = 0.5: The event is equally likely to happen or not happen
• 0.5 < P < 1: The event is likely to happen
• P = 1: The event is certain (will definitely happen)
Example 3: Interpreting Probabilities

Classify each probability on the scale:

• P(flipping heads on a fair coin) = 0.5 → Equally likely

• P(rolling a 7 on a standard die) = 0 → Impossible

• P(sun rising tomorrow) ≈ 1 → Certain

• P(drawing an ace from a deck) = 4/52 ≈ 0.077 → Unlikely

📝 Practice Question

The probability of an event is 0.85. How would you describe this event?
A) Impossible
B) Unlikely
C) Equally likely
D) Very likely

4. Complementary Probabilities

The complement of an event is everything that is NOT that event. The probabilities of an event and its complement always add up to 1.

P(Event) + P(NOT Event) = 1

Therefore: P(NOT Event) = 1 - P(Event)
Why This is Useful:
Sometimes it's easier to calculate the probability of something NOT happening and then subtract from 1. This is especially helpful when the complement has fewer outcomes to count!
Example 4: Using Complements

A weather forecast says there's a 30% chance of rain. What's the probability it WON'T rain?

Solution:

• P(rain) = 0.30

• P(no rain) = 1 - 0.30 = 0.70 or 70%

Example 5: Practical Application

In a class of 25 students, 18 passed the exam. What's the probability a randomly selected student did NOT pass?

Solution:

• P(passed) = 18/25 = 0.72

• P(did not pass) = 1 - 0.72 = 0.28 or 28%

Or we could calculate directly: 7 students didn't pass, so P = 7/25 = 0.28

📝 Practice Question

The probability that a randomly selected person owns a smartphone is 0.92. What is the probability that a randomly selected person does NOT own a smartphone?
A) 0.92
B) 0.08
C) 0.18
D) 1.92

5. Putting It All Together

📝 Challenge Question

A spinner is divided into 8 equal sections: 3 red, 2 blue, 2 yellow, and 1 green. What is the probability of NOT landing on red? Express your answer as a fraction in simplest form.
A) 3/8
B) 5/8
C) 1/2
D) 2/3

🎯 Lesson Summary

Congratulations! You've mastered calculating basic probabilities. Here's what you learned:

  • ✅ Basic probability formula: P(Event) = Favorable outcomes ÷ Total outcomes
  • ✅ Converting forms: Move between fractions, decimals, and percentages
  • ✅ Probability scale: All probabilities range from 0 (impossible) to 1 (certain)
  • ✅ Complementary events: P(Event) + P(NOT Event) = 1

Next lesson: We'll explore counting principles to handle more complex probability problems!