Lesson 55: Finding Probability

New Concept

For equally likely outcomes, theoretical probability is the ratio of favorable to total outcomes: $P(\operatorname{event}) = \frac{\operatorname{number\;of\;favorable\;outcomes}}{\operatorname{total\;number\;of\;outcomes}}$.

Why it matters

Probability is the mathematical tool for quantifying uncertainty, a skill essential for strategic decision-making in fields from finance to scientific research. Mastering it allows you to model risk and predict outcomes where others only see chance.

What’s next

Next, you'll apply this definition to calculate probabilities in scenarios involving geometric shapes, combinations, and both independent and dependent events.

For equally likely outcomes, the theoretical probability $P$ of an event is the ratio:

Example 1: $P(\text{rolling a number > 4 on a 6-sided die}) = \frac{2 \text{ (5, 6)}}{6 \text{ (1, 2, 3, 4, 5, 6)}} = \frac{1}{3}$.

Example 2: A random integer from 1 to 100 is chosen. $P(\text{multiple of 25}) = \frac{4 \text{ (25, 50, 75, 100)}}{100} = \frac{1}{25}$.

For independent events A and B, $P(A \text{ and } B) = P(A) \cdot P(B)$. For dependent events A and B, $P(A \text{ and } B) = P(A) \cdot P(B|A)$.

Example 1 (Independent): Rolling a die and flipping a coin. $P(\text{rolling a 6 and getting heads}) = \frac{1}{6} \cdot \frac{1}{2} = \frac{1}{12}$.

Example 2 (Dependent): A bag has 3 green and 5 yellow marbles. $P(\text{picking green, then yellow without replacement}) = \frac{3}{8} \cdot \frac{5}{7} = \frac{15}{56}$.

Odds compare favorable outcomes to unfavorable outcomes.

Example 1: A bag contains 4 green and 7 orange marbles. The odds in favor of picking green are $\frac{4}{7}$ or $4:7$.
Example 2: The odds in favor of an event are $3:8$. The probability of the event is $\frac{3}{3+8} = \frac{3}{11}$.

Forget total outcomes for a second! Odds are all about a direct showdown: the chances of winning versus the chances of losing. It’s a ratio that compares your desired outcomes directly to all the other, not-so-great outcomes. This is the language of sports betting and races, focusing purely on the favorable versus the unfavorable.

Conditional probability $P(B|A)$ is the probability of event $B$ given that event $A$ has occurred.

Example 1: From a bag with 2 red and 6 blue marbles, $P(\text{blue|red was picked first without replacement}) = \frac{6}{7}$.
Example 2: From a standard 52-card deck, $P(\text{King | Face Card}) = \frac{4 \text{ Kings}}{12 \text{ Face Cards}} = \frac{1}{3}$.
Example 3: When rolling a die, $P(\text{rolling a 2 | an even number was rolled}) = \frac{1}{3}$.

This is 'if-then' probability! It answers the question: 'What are the chances of B happening, if I already know A happened?' This new information shrinks the sample space, changing the odds. It is like asking the probability of drawing a Queen, given you already know the card you drew is a face card. It is all about updating your predictions.