Notes
Basics
- For equally likely outcomes, $\displaystyle P(A) = \frac{\text{outcomes in } A}{\text{total outcomes}}$. Probabilities are between 0 and 1.
- Complement: $P(A^c) = 1 - P(A)$.
- Addition rule: $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$. For mutually exclusive events the last term is 0.
Multiplication, conditional and independence
- Conditional: $\displaystyle P(A \mid B) = \frac{P(A \text{ and } B)}{P(B)}$.
- Multiplication rule: $P(A \text{ and } B) = P(B)\,P(A \mid B)$. If $A$ and $B$ are independent, $P(A \text{ and } B) = P(A)P(B)$ and $P(A \mid B) = P(A)$.
Tables and trees
- Two-way tables: divide by the grand total for joint and marginal probabilities, and by a row or column total for conditional probabilities.
- Without replacement, probabilities change after each draw.
Topics in this unit
Each topic has its own short notes and practice.
Practice
$1 - 0.12 = 0.88$.
$P(A \text{ and } B) = 0.6 + 0.5 - 0.8 = 0.3$.
Six of the 36 outcomes sum to 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1). $\displaystyle \frac{6}{36} = \frac{1}{6}$.
Restrict to the 80 band students: $\displaystyle \frac{20}{80} = 0.25$.
$0.2 \times 0.5 = 0.1 = P(A \text{ and } B)$, so they are independent. They are not mutually exclusive, since $P(A \text{ and } B) \ne 0$.
$\displaystyle \frac{13}{52} \cdot \frac{12}{51} = \frac{156}{2652} = \frac{1}{17}$.
Free response
Write your answer first, then open the scoring guide.
Free-response question
A school has 80 students in a language survey. Of the 45 juniors, 30 take Spanish. Of the 35 seniors, 20 take Spanish.
(a) Find $P(\text{Spanish})$ and $P(\text{junior and Spanish})$ for a randomly chosen student.
Show the answer and scoring guide
Spanish students: $30 + 20 = 50$, so $\displaystyle P(\text{Spanish}) = \frac{50}{80} = 0.625$. $\displaystyle P(\text{junior and Spanish}) = \frac{30}{80} = 0.375$.
Check off the points you earned:
(b) Find $P(\text{Spanish} \mid \text{junior})$.
Show the answer and scoring guide
$\displaystyle \frac{30}{45} = \frac{2}{3} \approx 0.667$.
Check off the points you earned:
(c) Are taking Spanish and being a junior independent? Justify.
Show the answer and scoring guide
No. $P(\text{Spanish} \mid \text{junior}) \approx 0.667$ is not equal to $P(\text{Spanish}) = 0.625$.
Check off the points you earned: