Basic Probability
1. Core Idea
For equally likely outcomes:
P(event) = (number of favourable outcomes) / (total number of outcomes)
Every probability lies between 0 and 1. A probability of 0 means impossible; 1 means certain.
The hardest part is usually counting the total correctly — which is why counting techniques come first.
2. Must-Know Rules
| Concept | Rule |
|---|---|
| Probability range | 0 <= P <= 1 |
| Certain event | P = 1 |
| Impossible event | P = 0 |
| Complement | P(not A) = 1 - P(A) |
| Sum of all outcomes | equals 1 |
| Odds in favour a : b | P = a/(a + b) |
The complement rule is the workhorse. Any question containing "at least one" should trigger it:
P(at least one) = 1 - P(none)
Standard setups and their totals:
| Experiment | Total outcomes |
|---|---|
| One coin | 2 |
| Two coins | 4 |
| n coins | 2^n |
| One die | 6 |
| Two dice | 36 |
| A standard deck | 52 cards, 4 suits of 13, 26 red, 26 black, 12 face cards |
Two-dice sums. The most likely sum is 7 (6 ways out of 36). Sums 2 and 12 have 1 way each. The count for a sum s (2 to 7) is s - 1, and it mirrors downward after 7.
3. Worked Examples
Example 1 — Simple count
A bag has 5 red, 3 blue and 4 green marbles. One is drawn at random. What is P(not blue)?
Total = 12. Blue = 3.
P(blue) = 3/12 = 1/4, so P(not blue) = 1 - 1/4 = 3/4.
Example 2 — Two dice
Two fair dice are rolled. What is the probability the sum is 9?
Favourable: (3,6), (4,5), (5,4), (6,3) -> 4 ways.
Total = 36.
P = 4/36 = 1/9.
Example 3 — At least one
Three fair coins are tossed. What is the probability of getting at least one head?
P(no heads) = P(all tails) = (1/2)^3 = 1/8.
P(at least one head) = 1 - 1/8 = 7/8.
Counting the cases directly would take four separate calculations; the complement takes one.
4. GRE Traps
- Wrong denominator. Two dice give 36 outcomes, not 12 or 11.
- Forgetting that (3,6) and (6,3) are different outcomes when the dice are distinguishable.
- Probability greater than 1. A sure sign of a counting error.
- Adding probabilities of non-mutually-exclusive events. P(A or B) = P(A) + P(B) - P(A and B).
- Assuming replacement. "Two marbles are drawn" usually means without replacement, so the second draw has a smaller total.
- Confusing odds with probability. Odds of 3 : 2 in favour means a probability of 3/5, not 3/2.
- Enumerating "at least one" case by case and missing a case.
5. Speed Tricks
- Count the total first. Get the denominator right and the rest usually follows.
- "At least one" -> 1 minus none. Automatic.
- Draw the 6x6 grid for two-dice questions once in class; students then recall the pattern rather than recounting.
- Simplify fractions as you go — GRE answer choices are in lowest terms.
- Sanity check the size: a "likely" event should give a probability well above 1/2.
- For sequential draws, multiply stage by stage, adjusting the counts each time.
6. Self-Check
Q1. A die is rolled. What is the probability of getting a number greater than 4?
Q2. A card is drawn from a standard deck. What is P(a face card)?
Q3. Two dice are rolled. What is the probability the sum is at least 11?
Answers
A1. Favourable: 5, 6 -> 2/6 = 1/3.
A2. 12 face cards out of 52 = 3/13.
A3. Sum 11: (5,6),(6,5) = 2 ways. Sum 12: (6,6) = 1 way. Total 3/36 = 1/12.
7. One-Line Summary for the Board
P = favourable over total. Get the total right first. "At least one" means 1 minus none.