Probability Basics: A Student Guide

Probability tells you how likely something is, on a scale from 0 (impossible) to 1 (certain). You find it by comparing the outcomes you care about to all the outcomes that could happen. This guide walks through simple probability, independent and dependent events, the complement rule, expected value, and counting, with worked examples you can repeat on your own.

Direct Answer

Probability of an event equals the number of favorable outcomes divided by the total number of equally likely outcomes. For two independent events, multiply their separate probabilities. For two dependent events, multiply too, but update the second probability after the first one happens. Khan Academy frames the same idea by starting with a coin flip, where two outcomes are equally likely, then building toward compound and conditional cases (Khan Academy, Probability: the basics).

Probability at a Glance

What Probability Means

The basic formula is:

P(event) = (number of favorable outcomes) / (total number of equally likely outcomes)

The phrase equally likely matters. A fair six sided die has six outcomes, each with the same chance. A loaded die does not, so the simple fraction breaks down.

Worked example. What is the probability of rolling a 4 on a fair die?

One favorable outcome out of six total, so 1/6, about 0.167, or 16.7 percent.

Worked example. What is the probability of rolling an even number (2, 4, or 6)?

Three favorable outcomes out of six, so 3/6 = 1/2 = 0.5, or 50 percent.

The same logic applies to any finite sample space where outcomes are equally likely. Pull one card from a well shuffled 52 card deck and the chance it is a heart is 13/52 = 1/4.

The Complement Rule

The complement of an event is everything else that could happen. Because all probabilities add to 1, P(not A) = 1 - P(A).

Worked example. The probability of not rolling a 4 on a die is 1 - 1/6 = 5/6.

This rule saves time when the opposite is easier to count. Suppose you want the probability that at least one student in a group of three was born on a weekday. Counting "at least one" directly is messy. Counting the complement, "all three born on the weekend," is simpler, then subtract from 1.

Independent Events

Two events are independent if the result of one does not change the chance of the other. For independent events, the probability that both happen is the product of their probabilities.

P(A and B) = P(A) × P(B)

Worked example. Flip a fair coin twice. What is the probability of heads then heads?

Each flip is 1/2, and they are independent, so (1/2) × (1/2) = 1/4, or 25 percent.

A common mistake here is the gambler's fallacy: thinking that after several heads, tails becomes "due." Each flip stays at 1/2. Khan Academy's lesson on compound probability of independent events shows this directly by listing all outcomes or by multiplying (Khan Academy, Compound probability of independent events).

Dependent Events

If the first outcome changes the second, the events are dependent. You adjust the second probability after the first occurs, a process called conditional probability.

Worked example. A bag has 3 red and 2 blue marbles. Draw one without replacing it, then draw again. What is the probability both are red?

First red: 3/5. Now 2 red remain out of 4 total, so second red: 2/4 = 1/2. Multiply: (3/5) × (1/2) = 3/10, or 30 percent.

Notice the denominator dropped from 5 to 4 because the first marble was not put back. If you had replaced it, the events would be independent again and the answer would be (3/5) × (3/5) = 9/25.

The Addition Rule for "Or"

For "A or B," add the individual probabilities, then subtract the overlap so you do not count it twice.

P(A or B) = P(A) + P(B) - P(A and B)

Worked example. A card is drawn from a standard 52 card deck. What is the probability it is a heart or a king?

Hearts: 13/52. Kings: 4/52. The king of hearts is counted in both, so subtract 1/52. Total = (13 + 4 - 1)/52 = 16/52 = 4/13, about 30.8 percent.

When two events cannot both happen, they are mutually exclusive and the overlap is zero, so you just add.

Expected Value

Expected value is the average result you would expect over many repeated trials. Multiply each outcome by its probability, then add the products.

Worked example. A game pays 10 dollars with probability 1/5 and pays nothing otherwise. The expected value is (10 × 1/5) + (0 × 4/5) = 2 dollars. Over many plays, you would average about 2 dollars per game.

Expected value helps you judge whether a deal is fair in the long run. It does not predict a single outcome. You might win 10 dollars on the first play or lose several times in a row. The average only shows up across many repetitions.

Counting Outcomes

When choices combine, multiply the counts. A shirt comes in 3 colors and 2 sizes, so 3 × 2 = 6 combinations. This multiplication rule underlies many probability counts and sits at the base of permutations and combinations, which you can explore in our statistics guide for students.

Worked example. A restaurant offers 4 main dishes, 3 sides, and 2 drinks. The number of full meals is 4 × 3 × 2 = 24.

When order matters, use permutations. When it does not, use combinations. For most high school problems, setting up the multiplication clearly beats memorizing formulas.

Common Misconceptions

• Assuming events are independent when the first one changes the second.
• Forgetting to subtract the overlap in "or" problems, which double counts.
• Using outcomes that are not equally likely, such as a weighted spinner treated as fair.
• Confusing probability with odds. Probability is favorable over total; odds are favorable over unfavorable.
• Believing a streak must end. Independent trials have no memory.

Conditional Probability in Plain Terms

Conditional probability asks how the chance of one event changes once you know another happened. The notation P(A given B) reads as the probability of A, given that B already occurred.

Worked example. You draw one card from a standard 52 card deck. What is the probability it is a king, given that it is a face card? There are 12 face cards, the jack, queen, and king in each of the 4 suits, and 4 of them are kings, so the answer is 4/12 = 1/3. Without the condition, the chance of a king is only 4/52. The condition shrank the sample space from 52 outcomes to 12.

This connects directly to dependent events. When you drew the marble without replacement earlier, each draw was conditional on the previous one. Tree diagrams make this visible: branch by the first outcome, then branch by the second, and multiply along each path to get the probability of a full sequence.

A practical test: if P(A given B) equals P(A), the events are independent, and you can multiply the plain probabilities. If the number changes, they are dependent, and you must use the conditional value. That single check tells you which formula applies before you start calculating.

Putting the Rules Together

Real problems often blend several rules. Suppose you roll a die and flip a coin. What is the probability of an even number and heads? The die and coin are independent, so multiply: P(even) = 3/6 = 1/2, P(heads) = 1/2, and the combined probability is 1/2 × 1/2 = 1/4.

Now suppose you draw two cards without replacement and want one king and one queen in any order. The first card could be the king (4/52) then the queen (4/51), or the queen (4/52) then the king (4/51). Add the two paths: (4/52 × 4/51) + (4/52 × 4/51) = about 0.012, or 1.2 percent. Blending the addition and multiplication rules is the skill that exam questions test.

Frequently Asked Questions

What does a probability of 1 mean?

It means the event is certain to happen. A probability of 0 means it is impossible.

What is an independent event?

An event whose outcome does not affect the probability of another event. Coin flips are the standard example.

How do I find "A and B" for independent events?

Multiply their probabilities: P(A) multiplied by P(B).

Why subtract in "A or B"?

To avoid counting the overlap, where both happen, twice.

What is the complement rule?

P(not A) = 1 - P(A). It is handy when the opposite is easier to count than the event itself.

When do students learn probability?

Basics often appear in grade 6 or 7, with deeper work in high school statistics and again in our algebra basics guide when sequences and functions enter the picture.

Sources

• Khan Academy, Probability: the basics
• Khan Academy, Compound probability of independent events
• Khan Academy, Probability library