Simple probability.
Probability of any simple event is favorable outcomes over total outcomes. The whole topic builds from this one ratio.
A weather app says "70% chance of rain." A casino game advertises "1-in-38 chance of hitting your number." A friend offers to flip a coin to decide lunch — "50/50," they shrug. Three different numbers, three different scenarios, but the math underneath is identical: count the outcomes that count as "yes" and divide by the total number of outcomes possible.
That single ratio — favorable outcomes over total outcomes — is the foundation of every probability calculation in this topic. Lessons 2 through 6 build progressively more sophisticated machinery on top of it: counting outcomes when there are millions of them (L2), combining events with OR and AND (L3), ordered vs unordered selections (L4), the long-run average payout of a process (L5), and comparing predictions to actual data (L6).
Everything in this lesson is built from one move: count what counts, count the total, divide.
Probability is favorable / total.
Examples: rolling a die has six equally-likely outcomes; the probability of rolling an even number is 3/6 = 1/2. Drawing a card from a standard 52-card deck has 52 equally-likely outcomes; the probability of drawing a heart is 13/52 = 1/4. The math doesn't care about the scenario — it only cares about the count of favorable outcomes and the count of total outcomes.
Left: the ALEKS Q1 scenario — eight cards, five favorable (P through T), three not, so P = 5/8. Right: every simple-event probability sits somewhere on the 0-to-1 number line, with 0 = impossible and 1 = certain.
The probability of an event A in a sample space of equally-likely outcomes is P(A) = (favorable outcomes) / (total outcomes), always a value in [0, 1].
Five words you'll see all week
- Experiment Any process with an uncertain outcome: rolling a die, flipping a coin, drawing a card, spinning a wheel. The probability questions in this topic are about experiments.
- Outcome A single possible result of the experiment. Rolling a die has 6 outcomes (1, 2, 3, 4, 5, 6). Drawing a card has 52.
- Sample space The set of all possible outcomes. For a die, the sample space is {1, 2, 3, 4, 5, 6}. For a card draw, the sample space has 52 elements.
- Event A subset of the sample space — the outcomes we're interested in counting as "yes." "Rolling an even number" is the event {2, 4, 6} inside the die's sample space.
- Equally likely Every outcome in the sample space has the same chance of occurring. This lesson assumes equally-likely outcomes (fair die, well-shuffled deck). When that assumption fails, the simple-ratio formula doesn't apply directly — you weight by individual outcome probabilities instead (Lesson 5).
- Probability range Every probability is a number between 0 and 1 (inclusive). 0 means impossible, 1 means certain, and everything in between is the rest of life. If you compute a probability that comes out negative or above 1, you made an arithmetic error somewhere.
Eight cards, five favorable.
"A box contains eight cards labeled P, Q, R, S, T, U, V, and W. One card will be randomly chosen. What is the probability of choosing a letter from P to T? Write your answer as a fraction."
Identify the sample space.
The sample space is the set of all eight cards: {P, Q, R, S, T, U, V, W}. Total outcomes = 8.
Count favorable outcomes.
The event "choosing a letter from P to T" includes the cards P, Q, R, S, T. That's five outcomes:
Build the ratio.
Divide favorable by total:
Check whether the fraction reduces.
5 and 8 share no common factors greater than 1 (5 is prime; 8 = 2×2×2). The fraction 5/8 is already in lowest terms. That's the final answer.
→ reduce when you canThree problems. Count, divide, reduce.
A fair six-sided die is rolled. What is P(even)? Write your answer as a fraction in lowest terms.
A standard 52-card deck is shuffled and one card is drawn. What is P(heart)? Write your answer as a fraction in lowest terms.
A raffle has 200 tickets sold. You bought 15 of them. What is the probability that your ticket wins the single grand prize? Fraction in lowest terms.
Three fast questions before you move on.
Q1. Which value cannot be the probability of any event?
Q2. A bag contains 4 red marbles, 6 blue marbles, and 10 green marbles. One marble is drawn at random. What is P(blue)?
Q3. If P(rain) = 0.7, what is P(no rain)?
From one ratio to every probability question.
Everything in this topic comes back to favorable / total. The next five lessons give you better tools for counting when the sample space is huge (L2, fundamental counting principle), for combining events with OR and AND (L3, addition rule), for ordered vs unordered selections (L4, permutations and combinations), for the long-run value of a process (L5, expected value), and for comparing predictions to data (L6, theoretical vs experimental).
Next: Lesson 2 introduces the fundamental counting principle — the move that handles "how many possible outcomes are there?" when listing the sample space by hand would take all day.
Continue to Lesson 02Different angle? Need another rep? These are optional — tap any that look helpful.
Probability explained — Independent and dependent events
Sal opens with the coin and die — favorable outcomes over total outcomes — and establishes the 0-to-1 range. The cleanest intro to P(event) for a first-time learner. Evergreen despite being older.
Probability with playing cards
Direct application of the favorable/total ratio to a standard 52-card deck — sets up the deck framework students will see again in Lesson 6.3.