Probability & Counting Principles (Sections 9.1–9.4)

Section 9.1 – Determining Probabilities

• Key vocabulary

  • Experiment = an activity that produces observable results (outcomes).

  • Outcome = each individual result of an experiment.

  • Sample space $S$ = set of all possible outcomes; outcomes in $S$ are mutually exclusive & collectively exhaustive.

  • Event = any subset of $S$.

• Modeling a sample space

  • Can be shown by sketches, tree-diagrams, or roster form ${\,}$.

  • Example: single coin toss → $S={H,T}$.

  • Die-rolling example

  • Sample space: $S={1,2,3,4,5,6}$ (drawing, tree diagram, set list).

  • Events defined

    • $A={5}$ (rolling a 5)

    • $B={2,4,6}$ (even number)

    • $C={2,3,5}$ (prime number)

Experimental (Empirical) Probability

• Based on data from actually performing the experiment.

• Expressed as fraction/decimal/percent of number of times event occurs over total trials.

• Accuracy increases as number of trials grows → Law of Large Numbers / Bernoulli’s Theorem (Theorem 9-1)

  • “As the number of trials $n$ increases, the experimental probability $\hat P(A)$ approaches the theoretical probability $P(A)$.”

Theoretical Probability (Equally Likely Outcomes)

• If all outcomes in $S$ are equally likely:
  

  $P(A)=\frac{n(A)}{n(S)}$

• Single fair coin → $P(H)=P(T)=\tfrac12$.

• Theorem 9-2: For any event $A$, $0\le P(A)\le1$.

Worked example (Choosing a random integer $1!\text{–}!25$)

• Given $S={1,2,\dots,25}$, all equally likely.

  • Even number (A): $n(A)=12$ → $P(A)=\tfrac{12}{25}$.

  • Number > 25 (B): $n(B)=0$ → $P(B)=0$.

  • Number < 26 (C): $n(C)=25$ → $P(C)=1$.

  • Prime (D): $n(D)=9$ → $P(D)=\tfrac{9}{25}$.

  • Even & prime (E): $n(E)=1$ (only 2) → $P(E)=\tfrac{1}{25}$.

Section 9.2 – Rules for Combined Events

Mutually Exclusive (Disjoint) Events

• Definition: $A \cap B = \varnothing$ (cannot occur together).

• Additive rule (Theorem 9-4) for mutually exclusive: $P(A\cup B)=P(A)+P(B).$

General Addition Rule

$P(A\cup B)=P(A)+P(B)-P(A\cap B).$

Independent Events

• Definition: Occurrence of $A$ has no influence on $B$ ⇒ $P(B|A)=P(B).$

• Multiplicative rule (Theorem 9-7): $P(A\cap B)=P(A)\,P(B).$

Dice Illustrations

• Rolling two fair dice (36 equally likely ordered pairs).

  • Event A: “double sixes” ⇒ $1/36$.

  • Event B: “sum 7 or 11” ⇒ $\tfrac{8}{36}=\tfrac{2}{9}$.

  • A & B share no common outcome ⇒ mutually exclusive.

• Sum even (E) vs sum prime (F)

  • $n(E)=18,\; n(F)=15,\; n(E\cap F)=1$ (only sum 2).

  • $P(E\cup F)=\tfrac{18}{36}+\tfrac{15}{36}-\tfrac{1}{36}=\tfrac{32}{36}=\tfrac{8}{9}.$

Sequential Sampling – With vs. Without Replacement

• Box with 11 letters, specific word “BABY” drawn in order.

  • Without replacement:
    

    $P(BABY)=\frac{2}{11}\cdot\frac{1}{10}\cdot\frac{2}{9}\cdot\frac{1}{8}=\frac{1}{3\,960}.$

  • With replacement:
    

    $P(BABY)=\left(\frac{2}{11}\right)^2\left(\frac{1}{11}\right)^2=\frac{4}{14\,641}.$

  • Tree diagrams illustrate successive conditional probabilities.

Section 9.3 – Applications: Odds & Expected Value

Odds

• For event $A$ with probability $P(A)$:

  • Odds in favor $=\dfrac{P(A)}{1-P(A)}$ → expressed as $m:n$ where $m=P(A),\;n=1-P(A)$ after scaling to integers.

  • Odds against $=\dfrac{1-P(A)}{P(A)}$ (reverse ratio).

• Equally likely outcomes version:

  • Odds in favor =\dfrac{\text{# favorable}}{\text{# unfavorable}}.

• Examples

  • Die, number < 5 ⇒ favorable 4, unfavorable 2 → odds 2:1.

  • Fair coin, heads ⇒ odds 1:1.

  • Ace from 52-card deck ⇒ odds 1:12.

  • Heart from deck ⇒ odds 1:3.

• Theorem 9-8: If odds in favor are $m:n$ then $P(E)=\dfrac{m}{m+n}$; if odds against are $m:n$ then $P(E)=\dfrac{n}{m+n}.$

Expected Value (Mathematical Expectation)

$E=\sum<em>{i=1}^{n} a</em>i\,P_i.$

• Fair game ⇒ (expected payoff) − (cost to play) $=0.$

• Coin-toss game example

  • Payoffs: $10$ (HH), $5$ (exactly one H), $0$ (TT).

  • Probabilities: $\tfrac14,\;\tfrac12,\;\tfrac14.$

  • $E=10\left(\tfrac14\right)+5\left(\tfrac12\right)+0\left(\tfrac14\right)=5$.

  • Cost $=5$ ⇒ net $=0$ → fair.

Dice-Sum Game (cost $5$)

• Payoff structure summarized:

  • Sum 2 or 12 → $10$ (prob $\tfrac{1+1}{36}=\tfrac{2}{36}$).

  • Sum 3 or 11 → $9$ (prob $\tfrac{2}{36}$).

  • Sum 4 or 10 → $8$ (prob $\tfrac{3+3}{36}=\tfrac{6}{36}$).

  • Sum 5 or 9 → $5$ (prob $\tfrac{4+4}{36}=\tfrac{8}{36}$).

  • Sum 6,7,8 → $0$ (prob $\tfrac{5+6+5}{36}=\tfrac{16}{36}$).

• Expected winnings (before cost): $E=10\left(\tfrac{2}{36}\right)+9\left(\tfrac{2}{36}\right)+8\left(\tfrac{6}{36}\right)+5\left(\tfrac{8}{36}\right)+0\left(\tfrac{16}{36}\right)=4$.

• Net expectation $=4-5=-1$ → unfair; favors the operator; average player loss $1 per game → rational decision: do not play.

Section 9.4 – Counting Techniques

Permutations (Ordered Arrangements, No Repetition)

• Number of permutations of $n$ objects taken $r$ at a time:
  

  $nP_r=\frac{n!}{(n-r)!}.$ (Order matters.)

• Illustrations

  • All 9 actors arranged in a line: $9P_9=9!=362\,880.$

  • Choosing 3 initials (no repeats) from 26 letters: $26P_3=26\times25\times24=15\,600.$

Combinations (Unordered Selections)

• Number of combinations of $n$ objects taken $r$ at a time:
  

  $\binom{n}{r}=nC<em>r=\frac{n!}{r!(n-r)!}=\frac{nP</em>r}{r!}.$

• Logic: count permutations then divide by $r!$ arrangements of each chosen group.

• Illustrations

  • Book-club chooses 3 of 42 books: $\binom{42}{3}=11\,480.$

  • Handshakes among 25 students (choose any 2): $\binom{25}{2}=300.$

Mixed Combination/Probability Example – Class Committee

• Class: 12 girls, 10 boys (total 22).
  

  a) Number of committees of 5 with 3 girls & 2 boys:
  

  $\binom{12}{3}\binom{10}{2}=9\,900.$
  

  b) Probability random 5-member committee has 3 G + 2 B:
  

  $\dfrac{\binom{12}{3}\binom{10}{2}}{\binom{22}{5}}=\dfrac{9\,900}{26\,334}=\tfrac{50}{133}\approx0.376.$
  

  c) Committees with no boys: $\binom{10}{0}\binom{12}{5}=792.$
  

  d) Probability of all-girl committee: $\dfrac{792}{26\,334}=\tfrac{4}{133}\approx0.030.$

Summary of Key Probability Properties (from text)

1. $P(\varnothing)=0$ (impossible event).

2. $P(S)=1$ (certain event).

3. For any event $A$, $0\le P(A)\le1$.

4. If events $A,B$ are mutually exclusive: $P(A\cup B)=P(A)+P(B).$

5. General addition: $P(A\cup B)=P(A)+P(B)-P(A\cap B).$

6. Complement rule: $P(A^c)=1-P(A).$

7. Independence: $P(A\cap B)=P(A)P(B).$

8. Odds–Probability conversion: if odds are $m:n$ in favor, $P=\tfrac{m}{m+n}$.

These principles link experimental data, theoretical models, and counting techniques, enabling systematic analysis of random events, fair-game design, and decision-making under uncertainty.