Investigation 8: Probability and Odds, Compound Events, Experimental Probability

New Concept

Odds show the ratio of favorable to unfavorable outcomes and are written with the word “to” or with a colon (:).

Why it matters

Understanding odds is your first step from just calculating chances to strategically comparing them, a key skill in everything from game theory to financial analysis. This concept introduces you to ratio-based thinking, a cornerstone of algebra for modeling relationships between quantities.

What’s next

Next, you’ll apply this to calculate the odds of drawing marbles from a bag and then explore even more complex compound events.

New Concept

Odds show the ratio of favorable to unfavorable outcomes and are written with the word “to” or with a colon (:).

Why it matters

Understanding odds is your first step from just calculating chances to strategically comparing them, a key skill in everything from game theory to financial analysis. This concept introduces you to ratio-based thinking, a cornerstone of algebra for modeling relationships between quantities.

What’s next

Next, you’ll apply this to calculate the odds of drawing marbles from a bag and then explore even more complex compound events.

Property

Probability is the ratio of favorable outcomes to the total number of possible outcomes. Odds are the ratio of favorable outcomes to unfavorable outcomes.

Examples

A bag has 2 red, 3 green, and 5 yellow marbles (10 total).
$P(\text{Green}) = \frac{3}{10}$
Odds of picking Green are 3 to 7, or 3:7.

Explanation

Think of a raffle! Probability is your ticket compared to all tickets sold. Odds are more personal—it’s the ratio of your winning tickets to all the losing tickets. This highlights your chances of winning against your chances of not winning, offering a different and exciting perspective on luck!

Property

To find the probability of two or more independent events occurring, multiply the probabilities of each event: $P(A \text{ and } B) = P(A) \cdot P(B)$.

Examples

What is the probability of flipping tails and rolling a 5 on a number cube?
$P(\text{Tails and 5}) = P(\text{Tails}) \cdot P(5) = \frac{1}{2} \cdot \frac{1}{6} = \frac{1}{12}$
What is the probability of flipping tails three times in a row?
$P(\text{Tails, Tails, Tails}) = \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{8}$

Explanation

This is like a video game combo! To land the full attack, you have to succeed at each button press. We multiply the probability of each individual move to find the chance of pulling off that epic final combo. Each event must happen for the compound event to succeed.

Property

Examples

A player made 60 free throws in 80 attempts. Her experimental probability of making a shot is $\frac{60}{80} = \frac{3}{4}$.
A spinner landed on blue 50 times out of 200 spins. The experimental probability is $\frac{50}{200} = \frac{1}{4}$.

Explanation

Forget theory—this is about what actually happens when you run an experiment. It's like finding a basketball player's free-throw percentage by looking at their real game stats, not just guessing what should happen. It is probability based on real-world data from repeated trials.