Chapter 26 — Statistics and Probability
Table of Contents
26.1 Statistics: Mean, Median, Mode
Measures of Central Tendency. Given a data set $x_1, x_2, \ldots, x_n$:
| Measure | Definition | |—|—| | Mean (average) | $\bar{x} = \dfrac{x_1 + x_2 + \cdots + x_n}{n}$ | | Median | Middle value when data is sorted. If $n$ is even, average the two middle values. | | Mode | Most frequently occurring value(s). | | Range | $\max - \min$ |
Example 26-1. Data: $3, 7, 7, 2, 9$.
- Mean: $(3 + 7 + 7 + 2 + 9)/5 = 28/5 = 5.6$
- Median: Sort → $2, 3, 7, 7, 9$. Middle value = $7$.
- Mode: $7$ (appears twice).
- Range: $9 - 2 = 7$.
Weighted Average. If values have different “weights” (importances):
\[\bar{x} = \frac{w_1 x_1 + w_2 x_2 + \cdots + w_n x_n}{w_1 + w_2 + \cdots + w_n}\]Example: A grade where homework (weight 1) = 90, midterm (weight 2) = 80, final (weight 3) = 70:
\(\text{Weighted avg} = \frac{1(90) + 2(80) + 3(70)}{1 + 2 + 3} = \frac{460}{6} \approx 76.7.\)
Geometric Mean. For positive values $x_1, x_2, \ldots, x_n$:
\[G = \sqrt[n]{x_1 \cdot x_2 \cdots x_n}\]Useful for rates of change and multiplicative processes.
Exercise 26-1. Find the mean, median, and mode of: $12, 15, 15, 18, 20, 22, 22, 22, 25$.
Exercise 26-2. If Gary’s test scores are $82, 90, 76, 88$, what score does he need on the 5th test to have an average of $85$?
Exercise 26-3. Explain how the mean can be misleading if a data set contains extreme outliers.
26.2 Basic Probability
Definition. The probability of an event $A$ is:
\[P(A) = \frac{\text{number of favorable outcomes}}{\text{total number of equally likely outcomes}}\]Always $0 \leq P(A) \leq 1$. $P(A) = 0$ means impossible; $P(A) = 1$ means certain.
Example 26-2. A fair 6-sided die is rolled. $P(\text{even}) = \frac{3}{6} = \frac{1}{2}$.
Example 26-3. Two fair coins are tossed. Outcomes: HH, HT, TH, TT.
\(P(\text{at least one head}) = \frac{3}{4}.\)
Complementary Probability.
\[P(\text{not } A) = 1 - P(A)\]Often it’s easier to compute $P(\text{not } A)$ and subtract from 1.
Exercise 26-4. A bag has 5 red, 3 blue, and 2 green marbles. What is the probability of drawing a blue marble?
Exercise 26-5. What is the probability of rolling a sum of 7 with two fair dice?
26.3 Probability Rules
Independent Events
Two events are independent if the outcome of one doesn’t affect the other. For independent events:
\(P(A \text{ and } B) = P(A) \cdot P(B)\)
Example 26-4. A coin is flipped 3 times. $P(\text{all heads}) = \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{8}$.
Mutually Exclusive Events
Events are mutually exclusive if they cannot both happen. For mutually exclusive events:
\[P(A \text{ or } B) = P(A) + P(B)\]For general (not necessarily exclusive) events:
\(P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)\)
Conditional Probability
The probability of $A$ given $B$ has occurred:
\[P(A \mid B) = \frac{P(A \text{ and } B)}{P(B)}\]If $A$ and $B$ are independent, then $P(A \mid B) = P(A)$.
Example 26-5. A bag has 5 red and 3 blue marbles. Two are drawn without replacement.
\[P(\text{both red}) = \frac{5}{8} \times \frac{4}{7} = \frac{20}{56} = \frac{5}{14}.\]The second draw depends on the first — these are not independent.
Warning. Do not multiply probabilities for events that are not independent unless you use conditional probability. Always ask: “Does the outcome of the first event affect the second?”
Exercise 26-6. A fair die is rolled twice. What is $P(\text{first roll} > \text{second roll})$?
Exercise 26-7. Cards are drawn without replacement from a standard 52-card deck. What is $P(\text{two aces in a row})$?
26.4 Counting and Probability
Many probability problems reduce to counting: $P(A) = \dfrac{|A|}{|S|}$ where $S$ is the sample space. Combine counting techniques from Chapter 25 with probability.
Example 26-6. Five cards are dealt from a standard deck. What is the probability of a flush (all same suit)?
Favorable: $\binom{4}{1}\binom{13}{5} = 4 \times 1287 = 5148$.
Total: $\binom{52}{5} = 2598960$.
\(P(\text{flush}) = \frac{5148}{2598960} \approx 0.00198.\)
Example 26-7 (Complementary Probability + Counting). What is the probability that among 5 people, at least two share a birthday? (Assume 365 days, uniform.)
Easier to compute $P(\text{no shared birthday})$:
\[P(\text{all different}) = \frac{365}{365} \cdot \frac{364}{365} \cdot \frac{363}{365} \cdot \frac{362}{365} \cdot \frac{361}{365} \approx 0.9729.\]\(P(\text{at least one match}) = 1 - 0.9729 \approx 0.0271.\)
Odds
Odds in favor of event $A$: the ratio of favorable to unfavorable outcomes.
\[\text{Odds} = \frac{P(A)}{P(\text{not } A)} = \frac{P(A)}{1 - P(A)}\]If $P(A) = \frac{3}{8}$, odds in favor are $3 : 5$.
Exercise 26-8. Three dice are rolled. What is the probability that all three show different numbers?
Exercise 26-9. A committee of 3 is chosen from 5 men and 4 women. What is the probability that the committee has at least one woman?
26.5 Expected Value
Definition. The expected value (or expectation) of a random variable $X$ is:
\[E[X] = \sum_{i} x_i \cdot P(X = x_i)\]It represents the “average” outcome over many trials.
Example 26-8. A fair die is rolled. What is the expected value?
\(E[X] = 1 \cdot \frac{1}{6} + 2 \cdot \frac{1}{6} + 3 \cdot \frac{1}{6} + 4 \cdot \frac{1}{6} + 5 \cdot \frac{1}{6} + 6 \cdot \frac{1}{6} = \frac{21}{6} = 3.5.\)
Example 26-9. A game costs $1 to play. You flip a coin: heads wins $3, tails wins $0. Expected value:
\[E[\text{profit}] = (3 - 1) \cdot \frac{1}{2} + (0 - 1) \cdot \frac{1}{2} = 1 - 0.5 = 0.5.\]On average, you gain $0.50 per game. This is a “favorable” game.
Linearity of Expectation. For any random variables $X$ and $Y$ (even if dependent!):
\[E[X + Y] = E[X] + E[Y]\]This is extremely powerful. Example: Expected number of heads in $n$ fair coin flips = $n \cdot \frac{1}{2} = \frac{n}{2}$.
Exercise 26-10. A lottery ticket costs $2. With probability $\frac{1}{1000}$ you win $500. What is the expected value of buying a ticket?
Exercise 26-11. Roll two dice and sum them. What is the expected sum?
The Big Picture — What Is Probability?
We’ve defined probability as $\dfrac{\text{favorable}}{\text{total}}$, assuming equally likely outcomes. But what if outcomes aren’t equally likely? What about “the probability it rains tomorrow”?
There are actually different interpretations of probability:
- Classical: Counting equally likely outcomes (what we’ve been doing).
- Frequentist: The long-run frequency of an event’s occurrence.
- Bayesian: A measure of belief or confidence.
Each has strengths and weaknesses. Probability theory developed over centuries and remains philosophically subtle.
Problems to Solve for Chapter 26
Problem 478. The average of 5 numbers is 6. A sixth number is added. Now the average is 7. What is the sixth number? (MATHCOUNTS 1988)
Problem 479. The average of 10 numbers is 85. When one number is removed, the average of the rest is 84. What number was removed? (MAΘ 1990)
Problem 480. Two fair dice are rolled. Find the probability that their sum is a perfect square. (MATHCOUNTS)
Problem 481. A bag has 4 red and 6 blue marbles. Two are drawn without replacement. Find $P(\text{both red})$.
Problem 482. What is the probability that a randomly chosen 2-digit number is divisible by 3?
Problem 483. Three fair coins are tossed. What is the probability of getting more heads than tails?
Problem 484. In a group of 30 students, 15 play basketball, 12 play soccer, and 5 play both. A student is chosen at random. Find $P(\text{basketball or soccer})$.
Problem 485. Two cards are drawn from a standard deck without replacement. What is the probability that both are hearts? (AHSME)
Problem 486. If you roll a fair die until you get a 6, what is the expected number of rolls? (Hint: geometric distribution.)
Problem 487. A class has 8 boys and 12 girls. Two students are randomly chosen for a project. What is the probability that both are boys?
Problem 488. Three people are randomly assigned to 3 rooms (one per room). What is the probability that person A gets room 1?
Problem 489. The probability of rain on Saturday is $0.4$ and on Sunday is $0.3$ (independent). What is the probability it rains on at least one day?
Problem 490. A student guesses randomly on a 5-question true/false quiz. What is the probability of getting at least 4 correct?
Problem 491. A box has 3 red, 2 blue, and 1 green ball. Balls are drawn one at a time without replacement. What is the probability that the red balls are drawn on the first 3 draws?
Problem 492. The median of ${2, 7, x, 15, 20}$ (already sorted for $x \geq 7$) is $x$. If the mean equals the median, find $x$. (MAΘ 1987)
Problem 493. Eleven members of a club each have an average score of 80. A twelfth member joins with a score of 92. What is the new average?
Problem 494. Two dice are rolled. Given that their sum is at least 8, what is the probability that the sum is 10? (Conditional probability.)
Problem 495. A drawer has 5 red socks and 7 blue socks. How many socks must be drawn to guarantee a matching pair?
Problem 496. In a game, you flip a coin. Heads: win $5. Tails: lose $3. What is the expected value per game?
Problem 497. If $P(A) = 0.6$ and $P(B) = 0.5$, what are the maximum and minimum possible values of $P(A \text{ and } B)$?
Problem 498. An urn contains 10 balls, numbered 1 to 10. Three balls are drawn (without replacement). What is the probability that the largest number drawn is 7? (AMC/AHSME)
Problem 499. Four fair coins are tossed. What is the expected number of heads?