Probability: Concepts, Rules, and Counting Methods

Probability: Concepts, Rules, and Counting Methods

Section 3.1: Basic Concepts of Probability

This section introduces foundational terms and concepts in probability, including the structure of probability experiments, sample spaces, events, and the calculation of probabilities.

• Probability Experiment: An action or trial that produces specific results, such as counts, measurements, or responses.

• Outcome: The result of a single trial in a probability experiment.

• Sample Space: The set of all possible outcomes of a probability experiment.

• Event: A subset of the sample space; may consist of one or more outcomes.

• Simple Event: An event that consists of a single outcome.

Example: Rolling one die produces a sample space: {1, 2, 3, 4, 5, 6}.

Example: Rolling a die and flipping a coin produces a sample space of 12 outcomes, such as (1, Heads), (1, Tails), ..., (6, Heads), (6, Tails).

• Fundamental Counting Principle: If one event can occur in m ways and a second event in n ways, the number of ways both can occur in sequence is m × n. This extends to any number of events.

Example: Choosing a car from 3 manufacturers, 2 sizes, and 4 colors: 3 × 2 × 4 = 24 possible combinations.

• Probability of an Event: $P(E) = \frac{\text{number of outcomes in Event E}}{\text{number of outcomes in the sample space}}$

• Probability Range: $0 \leq P(E) \leq 1$; probabilities ≤ 0.5 are considered unusual.

Example: Probability of rolling at least 5 on a six-sided die: Outcomes are 5 and 6, so $P(E) = \frac{2}{6} = \frac{1}{3}$.

Example: Probability of drawing a Jack from a standard deck: $P(E) = \frac{4}{52} = \frac{1}{13}$.

• Complementary Events: The set of all outcomes not included in event E, denoted as E'.

• Complement Rules:

  • $P(E') = 1 - P(E)$

  • $P(E) = 1 - P(E')$

  • $P(E) + P(E') = 1$

Example: Probability of not choosing an employee aged 25-34 from a sample of 1000:

• Employees aged 25-34: 366

• Probability: $P(E) = \frac{366}{1000} = 0.366$

• Complement: $P(E') = 1 - 0.366 = 0.634$

Probability Using a Tree Diagram: Tree diagrams help visualize all possible outcomes, especially when combining multiple events (e.g., tossing a coin and spinning a spinner).

Probability Using the Fundamental Counting Principle: For an 8-digit college ID (digits 0-9, repeated): $10^8$ possible IDs.

Section 3.2: Conditional Probability and the Multiplication Rule

This section covers conditional probability, the distinction between independent and dependent events, and the multiplication rule for calculating joint probabilities.

• Conditional Probability: The probability of event B occurring given that event A has already occurred, denoted $P(B|A)$.

• Formula: $P(B|A) = \frac{P(A \text{ and } B)}{P(A)}$

Example: Probability that the second card is a queen, given the first card is a king (without replacement):

• After removing a king, 51 cards remain, 4 are queens.

• $P(B|A) = \frac{4}{51}$

Conditional Probability Using a Table: Given a table of children’s IQ and gene presence:

	Gene Present	Gene Not Present	Total
High IQ	33	19	52
Normal IQ	39	11	50
Total	72	30	102

Probability a child has high IQ given the gene: $P(\text{High IQ} | \text{Gene Present}) = \frac{33}{72}$

• Independent Events: Occurrence of one event does not affect the probability of the other. $P(B|A) = P(B)$ or $P(A|B) = P(A)$.

• Dependent Events: Occurrence of one event affects the probability of the other.

Multiplication Rule:

• If A and B are independent: $P(A \text{ and } B) = P(A) \cdot P(B)$

• If A and B are dependent: $P(A \text{ and } B) = P(A) \cdot P(B|A)$

Example: Probability of three successful knee surgeries (each with probability 0.85): $P = 0.85^3 = 0.614$

Example: Probability that none are successful: $P = (1 - 0.85)^3 = 0.003375$

Example: Probability that at least one is successful: $P = 1 - (1 - 0.85)^3 = 0.996625$

Example: Probability a medical student is matched to a residency and it is one of their top two choices: $P(\text{matched}) = 0.93$, $P(\text{top two} | \text{matched}) = 0.74$, so $P = 0.93 \times 0.74 = 0.6882$

Section 3.3: Addition Rule

This section explains how to calculate the probability of the union of two events, distinguishing between mutually exclusive and non-mutually exclusive events.

• Mutually Exclusive Events: Events that cannot occur at the same time. $P(A \text{ and } B) = 0$

• Addition Rule for Mutually Exclusive Events: $P(A \text{ or } B) = P(A) + P(B)$

• Addition Rule for Non-Mutually Exclusive Events: $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$

Example: Probability of selecting a card that is a 4 or an ace: $P(4) = \frac{4}{52}$, $P(\text{ace}) = \frac{4}{52}$, $P(4 \text{ and ace}) = 0$, so $P = \frac{8}{52} = \frac{2}{13}$

Example: Probability of rolling a number less than 3 or an odd number on a die:

• Numbers less than 3: 1, 2

• Odd numbers: 1, 3, 5

• Overlap: 1

• $P = \frac{2}{6} + \frac{3}{6} - \frac{1}{6} = \frac{4}{6} = \frac{2}{3}$

Example: Probability a blood donor has type O or type A:

	Type O	Type A	Type B	Type AB	Total
Rh- Positive	156	139	37	12	344
Rh- Negative	28	25	8	4	65
Total	184	164	45	16	409

Probability: $P = \frac{184 + 164}{409} = \frac{348}{409}$

Example: Probability a donor has type B or is Rh-negative:

• Type B: 45

• Rh-negative: 65

• Type B and Rh-negative: 8

• $P = \frac{45}{409} + \frac{65}{409} - \frac{8}{409} = \frac{102}{409}$

Section 3.4: Additional Topics in Probability and Counting

This section covers permutations, combinations, and their application in probability calculations.

• Permutation: An ordered arrangement of n objects. $n! = n \times (n-1) \times \ldots \times 1$

• Permutation Formula: $nPr = \frac{n!}{(n-r)!}$

• Distinguishable Permutations: $\frac{n!}{n_1! n_2! \ldots n_x!}$

• Combination: Selection of items without regard to order. $nCr = \frac{n!}{(n-r)! r!}$

Example: Number of ways to form a four-digit code with no repeated digits: $10 \times 9 \times 8 \times 7 = 5040$

Example: Number of ways to pick first, second, and third activity from 8: $8 \times 7 \times 6 = 336$

Example: Number of ways to form a three-person committee from 20 employees: $\frac{20!}{17! 3!} = 1140$

Example: Probability of winning the Mega Millions lottery (choose 5 numbers from 1-56, one Mega Ball from 1-46):

• Number of ways to choose 5 numbers: $\frac{56!}{51! 5!}$

• Number of ways to choose Mega Ball: 46

• Total possible tickets: $\frac{56!}{51! 5!} \times 46$

• Probability of winning: $\frac{1}{\frac{56!}{51! 5!} \times 46}$

Example: Probability that exactly one of four randomly selected corn kernels contains a dangerously high toxin level (from a sample of 400, 3 are dangerous):

• Number of ways to choose 1 dangerous kernel: $\binom{3}{1}$

• Number of ways to choose 3 safe kernels: $\binom{397}{3}$

• Total ways to choose 4 kernels: $\binom{400}{4}$

• Probability: $P = \frac{\binom{3}{1} \times \binom{397}{3}}{\binom{400}{4}}$