Probability measures how likely an event is, on a scale from 0 (impossible) to 1 (certain). A probability of 0.25 means the event happens about one time in four over many tries. You can write it as a fraction, a decimal or a percent.
The basic formula
When every outcome is equally likely:
P(event) = number of favorable outcomes / total number of outcomes
Example Roll a standard die. Find P(even) and P(greater than 4).
- Even outcomes are 2, 4, 6: P = 3/6 = 1/2.
- Greater than 4 means 5 or 6: P = 2/6 = 1/3.
Complements
The complement of an event is everything else. P(not A) = 1 − P(A).
The complement is the shortcut for any question that says “at least one.”
Example Roll a die 4 times. What is the probability of at least one 6?
- Listing every way to get at least one 6 is messy. The complement is “no 6 at all.”
- P(no 6 on one roll) = 5/6, so P(no 6 in four rolls) = (5/6)⁴ = 625/1296.
- P(at least one 6) = 1 − 625/1296 = 671/1296 ≈ 0.518.
“Or”: the addition rule
P(A or B) = P(A) + P(B) − P(A and B)
Subtract the overlap so it isn’t counted twice. If the events can’t happen together, the overlap is zero.
Example Draw one card from a standard deck. What is P(heart or king)?
13 hearts plus 4 kings, minus the king of hearts counted twice: 13/52 + 4/52 − 1/52 = 16/52 = 4/13.
“And”: the multiplication rule
For independent events, where one doesn’t affect the other: P(A and B) = P(A) · P(B).
For dependent events, multiply by the probability of the second event given that the first happened.
Example Draw two cards without replacement. What is the probability both are aces?
- First card: 4/52.
- Second card, with one ace and one card gone: 3/51.
- Multiply: 4/52 × 3/51 = 12/2652 = 1/221.
Independence isn’t the default
Drawing without replacement, choosing people for a team, or picking from a bag you don’t refill all create dependent events. Ask whether the first outcome changes the second before you multiply.
Conditional probability and two-way tables
P(A | B) means “the probability of A, given that B happened.” With a table, you restrict your attention to the row or column for B, and work only inside it.
Here is a survey of 200 students:
| Plays a sport | No sport | Total | |
|---|---|---|---|
| Grade 9 | 45 | 55 | 100 |
| Grade 10 | 60 | 40 | 100 |
| Total | 105 | 95 | 200 |
Example Find P(plays a sport | grade 10) and P(grade 10 | plays a sport)
- Given grade 10, look only at that row: 60 of 100, so 0.6.
- Given plays a sport, look only at that column: 60 of 105, so 4/7 ≈ 0.571.
Those two answers are different, and the SAT likes to test exactly that. Read which group comes after the word “given,” or after phrases like “of the students who,” and use that total as the denominator.
Counting: permutations and combinations
When outcomes are too many to list, count them.
- Permutations count arrangements where order matters: P(n, r) = n! / (n − r)!
- Combinations count selections where order doesn’t matter: C(n, r) = n! / (r!(n − r)!)
Example Three counting questions
- Arrange 5 books on a shelf: 5! = 120 ways.
- Choose a president, vice president and secretary from 10 people (the roles differ, so order matters): P(10, 3) = 10 × 9 × 8 = 720.
- Choose a committee of 3 from 10 people (the members are equal, so order doesn’t matter): C(10, 3) = 720/6 = 120.
Expected value
Expected value is the average result per try over many tries. Multiply each outcome by its probability and add.
Example A game costs $2. You roll a die and win $10 on a 6. Is it worth playing?
Expected winnings per game: (1/6)(10) = $1.67. Subtract the $2 cost: about −$0.33. On average you lose 33 cents a game.
Mistakes to avoid
- The gambler’s fallacy. A coin that landed heads five times in a row is still 50/50 on the next flip.
- Adding when you should multiply. “And” for independent events means multiply.
- Forgetting the overlap in “or” problems.
Practice problems
A bag has 5 red, 3 blue and 2 green marbles. Find P(blue).
Show answer
3/10.
From the same bag, find P(not green).
Show answer
4/5. 1 − 2/10 = 8/10.
Roll two dice. What is the probability their sum is 7?
Show answer
1/6. Six of the 36 equally likely outcomes add to 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1).
From the marble bag, draw two marbles without replacement. What is P(both red)?
Show answer
2/9. 5/10 × 4/9 = 20/90.
How many committees of 4 can be chosen from 9 people?
Show answer
126. C(9, 4) = (9 × 8 × 7 × 6)/(4 × 3 × 2 × 1).
P(A) = 0.5 and P(B) = 0.4, and A and B are independent. Find P(A and B) and P(A or B).
Show answer
0.2 and 0.7. P(A or B) = 0.5 + 0.4 − 0.2.
Practice
Questions students ask
What’s the difference between probability and odds?
Probability compares favorable outcomes to all outcomes. Odds compare favorable to unfavorable. Rolling a 6 has probability 1/6 and odds of 1 to 5.
Can a probability be greater than 1?
No. If you get a probability above 1 or below 0, there’s an error in the setup.
Where does probability show up besides games?
Weather forecasts, medical test results, insurance, and any study that uses a sample. For summarizing data, see Mean, Median, Mode and Standard Deviation.