AP Precalculus: Probability
Master probability formulas, counting techniques, and probability rules
📚 Understanding Probability
Probability measures the likelihood of an event occurring, expressed as a number between 0 (impossible) and 1 (certain). AP Precalculus covers fundamental probability concepts including counting techniques, conditional probability, and probability rules for combining events.
1 Basic Probability
The probability of an event A is the ratio of favorable outcomes to total possible outcomes in the sample space.
Problem: A bag contains 3 red, 5 blue, and 2 green marbles. Find P(blue).
Solution: \(P(\text{blue}) = \frac{5}{3+5+2} = \frac{5}{10} = 0.5\)
2 Permutations & Combinations
Permutations count arrangements where order matters. Combinations count selections where order doesn't matter.
How many ways can 3 people finish 1st, 2nd, 3rd from 8 runners?
\(P(8, 3) = \frac{8!}{5!} = 8 \times 7 \times 6 = 336\) ways
How many ways can a committee of 3 be chosen from 8 people?
\(C(8, 3) = \frac{8!}{3! \cdot 5!} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = \frac{336}{6} = 56\) ways
Permutation: rankings, arrangements, passwords. Combination: committees, teams, selections.
3 Probability with Counting Techniques
Use permutations or combinations to count favorable outcomes and total outcomes, then apply the basic probability formula.
Problem: From a deck of 52 cards, find P(drawing 2 aces when selecting 2 cards).
Favorable: \(C(4, 2) = 6\) ways to choose 2 aces from 4
Total: \(C(52, 2) = 1326\) ways to choose any 2 cards
Probability: \(P = \frac{6}{1326} = \frac{1}{221} \approx 0.0045\)
4 Two-Way Frequency Tables
A two-way frequency table displays data for two categorical variables. Use it to find joint, marginal, and conditional probabilities.
| Likes Math | Doesn't Like Math | Total | |
|---|---|---|---|
| Boys | 30 | 20 | 50 |
| Girls | 25 | 25 | 50 |
| Total | 55 | 45 | 100 |
Joint Probability
\(P(\text{Boy AND Likes Math}) = \frac{30}{100} = 0.30\)
Marginal Probability
\(P(\text{Likes Math}) = \frac{55}{100} = 0.55\)
Conditional Probability
\(P(\text{Likes Math | Boy}) = \frac{30}{50} = 0.60\)
5 Independent Events
Two events are independent if the occurrence of one does not affect the probability of the other.
Problem: A coin is flipped twice. Find P(heads on both).
Solution: Flips are independent. \(P(H_1 \cap H_2) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}\)
Independent events can both occur. Mutually exclusive events cannot both occur (if one happens, the other can't).
6 Conditional Probability
Conditional probability is the probability of event A occurring given that event B has already occurred.
Read as "probability of A given B"
Rearranged: Find P(A ∩ B)
\(P(A \cap B) = P(A|B) \cdot P(B)\)
Also Written As
\(P(A \cap B) = P(B|A) \cdot P(A)\)
Problem: \(P(\text{rain}) = 0.3\), \(P(\text{traffic jam | rain}) = 0.6\). Find P(rain and traffic jam).
Solution: \(P(\text{rain} \cap \text{jam}) = P(\text{jam | rain}) \times P(\text{rain}) = 0.6 \times 0.3 = 0.18\)
7 Addition Rule
The addition rule finds the probability that at least one of two events occurs (A or B or both).
Problem: \(P(\text{rain}) = 0.4\), \(P(\text{cold}) = 0.3\), \(P(\text{rain and cold}) = 0.1\). Find P(rain or cold).
Solution: \(P(\text{rain} \cup \text{cold}) = 0.4 + 0.3 - 0.1 = 0.6\)
Addition rule: P(A or B). Multiplication: P(A and B). Watch for these keywords in problems!
8 Multiplication Rule
The multiplication rule finds the probability that both events A and B occur.
Problem: Draw 2 cards without replacement. Find P(both red).
Solution: \(P = \frac{26}{52} \times \frac{25}{51} = \frac{650}{2652} = \frac{25}{102} \approx 0.245\)
📋 Quick Reference
Basic Probability
\(P(A) = \frac{\text{favorable}}{\text{total}}\)
Complement
\(P(A') = 1 - P(A)\)
Permutation
\(P(n,r) = \frac{n!}{(n-r)!}\)
Combination
\(C(n,r) = \frac{n!}{r!(n-r)!}\)
Conditional
\(P(A|B) = \frac{P(A \cap B)}{P(B)}\)
Addition Rule
\(P(A \cup B) = P(A) + P(B) - P(A \cap B)\)