Notes
Counting
- Counting principle: multiply the number of choices at each step.
- Permutations (order matters): $\displaystyle _nP_r = \frac{n!}{(n - r)!}$.
- Combinations (order doesn't matter): $\displaystyle _nC_r = \frac{n!}{r!\,(n - r)!}$.
- Divide by $k!$ for each group of $k$ identical items when arranging.
Probability rules
| Rule | Formula |
|---|---|
| Complement | $P(\text{not } A) = 1 - P(A)$ |
| Or | $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$ |
| And (independent) | $P(A \text{ and } B) = P(A)P(B)$ |
| And (dependent) | $P(A \text{ and } B) = P(A)P(B \mid A)$ |
| Conditional | $\displaystyle P(B \mid A) = \frac{P(A \text{ and } B)}{P(A)}$ |
Tips
- "At least one" problems are easiest with the complement.
- Without replacement, the totals shrink after each draw.
- In a two-way table, the "given" tells you which row or column to use as the denominator.
Topics in this unit
Each topic has its own short notes and practice.
Practice
$_7C_2 \cdot {_5C_1} = 21 \cdot 5 = 105$.
$6! = 720$.
$\displaystyle \frac{4}{52} \cdot \frac{3}{51} = \frac{12}{2652} = \frac{1}{221}$.
$\displaystyle _{30}C_5 = \frac{30 \cdot 29 \cdot 28 \cdot 27 \cdot 26}{120} = 142506$.
$P(\text{rain}) \cdot P(\text{late} \mid \text{rain}) = 0.3 \cdot 0.5$.
$0.6 + 0.3 - 0.6(0.3) = 0.9 - 0.18 = 0.72$.
Free response
Write your answer first, then open the scoring guide.
Free-response question
A screening test is given to 1,000 people. 20 of them have a certain condition, and 19 of those test positive. Of the 980 people without the condition, 98 test positive.
(a) What is the probability that a randomly chosen person tests positive?
Show the answer and scoring guide
Positives: $19 + 98 = 117$. $\displaystyle P(\text{positive}) = \frac{117}{1000} = 0.117$.
Check off the points you earned:
(b) A person tests positive. What is the probability they have the condition?
Show the answer and scoring guide
$\displaystyle P(\text{condition} \mid \text{positive}) = \frac{19}{117} \approx 0.162$.
Check off the points you earned:
(c) Are "testing positive" and "having the condition" independent? Explain with probabilities.
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$\displaystyle P(\text{positive} \mid \text{condition}) = \frac{19}{20} = 0.95$, but $P(\text{positive}) = 0.117$. These are not equal, so the events are not independent.
Check off the points you earned: