Basic probability, addition & multiplication rules, conditional probability, counting principles, and binomial probability — with worked examples.
Example
A bag has 3 red, 5 blue, and 2 green marbles. P(red) = 3/10 = 0.3
The complement A' ("A prime" or "not A") is everything that is NOT event A. Since all probabilities sum to 1, P(A) + P(A') = 1.
Example
P(rain tomorrow) = 0.35 → P(no rain) = 1 − 0.35 = 0.65
Mutually Exclusive Events
(Events that cannot happen at the same time)
P(A or B) = P(A) + P(B)
Example: rolling a 2 or a 5 on a die. Can't get both at once. P(2 or 5) = 1/6 + 1/6 = 1/3
Not Mutually Exclusive
(Events that can overlap — subtract the overlap)
P(A or B) = P(A) + P(B) − P(A and B)
Example: drawing a red card or a face card from a deck. Some face cards are red — subtract the overlap to avoid double-counting.
Independent Events
(One event does not affect the other)
P(A and B) = P(A) × P(B)
Example: flipping heads and then rolling a 4. P = 1/2 × 1/6 = 1/12
Dependent Events
(First event changes the probability of the second)
P(A and B) = P(A) × P(B|A)
P(B|A) = "probability of B given A already happened"
Read P(B|A) as "the probability of B given that A has already occurred." You are restricting the sample space to only outcomes where A happened.
Example
In a class: 30% study math (M), 20% study both math and science (M∩S).
P(S|M) = P(M and S) / P(M) = 0.20 / 0.30 ≈ 0.667
Among math students, about 67% also study science.
Fundamental Counting Principle
m × n (× p × ...)
If event 1 has m outcomes and event 2 has n outcomes, together there are m × n outcomes.
3 shirts × 4 pants = 12 outfits
Permutations (order matters)
P(n,r) = n! / (n−r)!
Arrangements of r items chosen from n. Order is important — ABC ≠ BAC.
P(5,3) = 5!/2! = 60 arrangements
Combinations (order doesn't matter)
C(n,r) = n! / [r!(n−r)!]
Selections of r items from n. Order is irrelevant — ABC = BAC = CAB.
C(5,3) = 5!/(3!·2!) = 10 groups
P(X = k) = C(n,k) · p^k · (1−p)^(n−k)
| Variable | Meaning |
|---|---|
| n | Total number of trials |
| k | Number of successes you want |
| p | Probability of success on one trial |
| 1−p | Probability of failure on one trial |
| C(n,k) | Number of ways to arrange k successes among n trials |
Use binomial probability when: there are exactly two outcomes (success/failure), each trial is independent, and n and p are constant throughout.
What is the probability of drawing two aces in a row from a standard 52-card deck?
P(1st ace) = 4/52 = 1/13
P(2nd ace | 1st was ace) = 3/51 = 1/17
P(both aces) = P(A) × P(B|A) = 4/52 × 3/51
= 12/2652 = 1/221 ≈ 0.0045
Without replacement → dependent. After the first ace is drawn, only 3 aces remain in a 51-card deck.
What is the probability of rolling a 6 on the first die and an even number on the second?
P(6 on die 1) = 1/6
P(even on die 2) = 3/6 = 1/2
Die rolls are independent → multiply
P(both) = 1/6 × 1/2 = 1/12 ≈ 0.083
A fair coin is flipped 8 times. What is P(exactly 3 heads)?
n = 8, k = 3, p = 0.5, (1−p) = 0.5
P(X = 3) = C(8,3) · (0.5)³ · (0.5)⁵
C(8,3) = 8!/(3!·5!) = 56
= 56 · 0.125 · 0.03125
= 56 · 0.00390625
= 0.21875 ≈ 21.9%
How many ways can a committee of 4 be chosen from a group of 10 people?
Order doesn't matter (a committee is a committee) → use combinations
C(10,4) = 10! / [4! · 6!]
= (10 × 9 × 8 × 7) / (4 × 3 × 2 × 1)
= 5040 / 24
= 210 different committees
| Situation | Rule | Formula |
|---|---|---|
| P(A or B), can't overlap | Addition — mutually exclusive | P(A) + P(B) |
| P(A or B), can overlap | Addition — general | P(A) + P(B) − P(A∩B) |
| P(A and B), independent | Multiplication — independent | P(A) × P(B) |
| P(A and B), dependent | Multiplication — conditional | P(A) × P(B|A) |
| P(B given A happened) | Conditional probability | P(A∩B) / P(A) |
| Arrangements (order matters) | Permutations | n!/(n−r)! |
| Selections (order irrelevant) | Combinations | n!/[r!(n−r)!] |
| Repeated trials, count successes | Binomial probability | C(n,k)·p^k·(1−p)^(n−k) |
Probability measures how likely an event is to occur. P(A) = (number of favorable outcomes) / (total number of outcomes). Probability always falls between 0 and 1 — where 0 means impossible and 1 means certain. For example, rolling a 3 on a standard die: P(3) = 1/6 ≈ 0.167.
Independent events: the outcome of one event does not affect the other. P(A and B) = P(A) × P(B). Example: flipping a coin twice — each flip is independent. Dependent events: the outcome of one event affects the probability of the other. P(A and B) = P(A) × P(B|A). Example: drawing two cards without replacement — after removing the first card, the deck changes, so the second draw is dependent on the first.
Permutations count arrangements where order matters: P(n,r) = n!/(n−r)!. Example: the number of ways to arrange 3 books chosen from 7 is P(7,3) = 7!/4! = 210. Combinations count selections where order does NOT matter: C(n,r) = n!/[r!(n−r)!]. Example: the number of ways to choose a committee of 3 from 7 people is C(7,3) = 35. A quick test: if swapping two items gives a different result, use permutations; if not, use combinations.
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