Probability and Relative Frequency: Foundations

[Video: Introduction to Probability and Relative Frequency]

Learn the fundamentals of probability and relative frequency, how to calculate them, and how they're used to make predictions about events.

Understanding Probability

Probability measures how likely an event is to occur, expressed as a number between 0 (impossible) and 1 (certain).

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

Examples of Probability

Probability of rolling a 3 on a fair die: 1/6 ≈ 0.167
Probability of drawing a heart from a standard deck: 13/52 = 1/4
Probability of flipping two heads in a row: 1/4

Understanding Relative Frequency

Relative frequency is the fraction or proportion of times an outcome occurs in an experiment or study.

Relative Frequency Formula: Relative Frequency = Number of times event occurs / Total number of trials

Theoretical Probability

Based on possible outcomes
What should happen
Used before conducting experiments
Example: Coin flip = 0.5 heads

Experimental Probability (Relative Frequency)

Based on actual results
What did happen
Used after collecting data
Example: 47 heads in 100 flips = 0.47

Probability Rules

Rule	Description	Example
Range	0 ≤ P(A) ≤ 1	Probability is always between 0 and 1
Complement	P(not A) = 1 - P(A)	P(not rain) = 1 - P(rain)
Independent Events	P(A and B) = P(A) × P(B)	P(heads and 6) = 0.5 × 1/6
Mutually Exclusive	P(A or B) = P(A) + P(B)	P(1 or 2 on die) = 1/6 + 1/6

Key Takeaways

Probability predicts likelihood before an event, relative frequency describes what happened after
As the number of trials increases, experimental probability approaches theoretical probability
Always check if events are independent or mutually exclusive when combining probabilities
Convert probabilities to percentages by multiplying by 100% when needed

Practice Question

A restaurant collected data on 200 customers' dessert orders:

Dessert	Number Ordered
Chocolate Cake	85
Ice Cream	60
Fruit Salad	35
Cheesecake	20

Based on this data, what is the relative frequency of customers who ordered either ice cream or cheesecake? Round to the nearest hundredth.

A) 0.30

B) 0.40

C) 0.45

D) 0.50

To find the relative frequency:

Add the number of customers who ordered ice cream and cheesecake
Divide this sum by the total number of customers (200)
Round the result to two decimal places

Step 1: Calculate total for ice cream and cheesecake: 60 (ice cream) + 20 (cheesecake) = 80

Step 2: Divide by total customers: 80 ÷ 200 = 0.40

Step 3: The relative frequency is already to two decimal places: 0.40

The correct answer is B) 0.40.

Note: Relative frequency is equivalent to probability in this context since we're using observed data to estimate probability.