Probability and Counting Rules: Fundamental Concepts and Applications

Section 3.1 - Basic Concepts of Probability and Counting

Probability Experiments and Outcomes

• Probability Experiment: An action or trial where specific results are obtained. Each repetition of the experiment is called a trial.

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

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

• Event: A subset of the sample space, consisting of one or more outcomes.

Tree Diagrams

• A tree diagram is a graphical representation that starts at a single point and branches out to show all possible outcomes of an experiment.

The Fundamental Counting Principle

• If one event can occur in m ways and a second event can occur in n ways, the number of ways the two events can occur in sequence is m × n.

• Example: A car's security code consists of four digits (0–9):

  • Each digit used only once: $10 \times 9 \times 8 \times 7 = 5040$ possible codes

  • Each digit can be repeated: $10 \times 10 \times 10 \times 10 = 10,000$ possible codes

  • Each digit can be repeated, but first digit cannot be 0 or 1: $8 \times 10 \times 10 \times 10 = 8,000$ possible codes

Classical (Theoretical) Probability

• Assumes all outcomes in the sample space are equally likely.

• Formula:

  $P(E) = \frac{\text{Number of outcomes in event } E}{\text{Number of outcomes in sample space}}$

• Examples:

  • Rolling a 3 on a six-sided die: $P(3) = \frac{1}{6} = 0.167$

  • Rolling a 7 on a six-sided die: $P(7) = \frac{0}{6} = 0$

  • Rolling a number less than 5: $P(<5) = \frac{4}{6} = 0.667$

Empirical (Statistical) Probability

• Based on observations from actual experiments.

• Formula:

  $P(E) = \frac{\text{Frequency of event } E}{\text{Total frequency}} = \frac{f}{n}$

  where $n = \Sigma f$

Subjective Probability

• Based on intuition, educated guesses, or estimates rather than exact calculations.

Range of Probabilities Rule

• The probability of any event E is between 0 and 1, inclusive.

• Formula:

  $0 \leq P(E) \leq 1$

• Interpretation:

  Probability	Interpretation
  0	Impossible
  0.25	Unlikely
  0.5	Even Chance
  0.75	Likely
  1	Certain

Complement of an Event

• The complement of event E (denoted E') is the set of all outcomes in the sample space that are not in E.

• Formulas:

  $P(E) + P(E') = 1$

  $P(E) = 1 - P(E')$

  $P(E') = 1 - P(E)$

Section 3.2 - Conditional Probability and the Multiplication Rule

Conditional Probability

• The probability of an event occurring, given that another event has already occurred.

• Notation: $P(B|A)$ ("probability of B, given A")

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

  • There are 51 cards left, 4 of which are queens.

  • $P(\text{2nd card is queen} | \text{1st card is king}) = \frac{4}{51} = 0.078$

Independent and Dependent Events

• Independent Events: The occurrence of one event does not affect the probability of the other.

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

• For independent events:

  $P(B|A) = P(B)$ or $P(A|B) = P(A)$

Multiplication Rule

• The probability that two events A and B both occur is:

  $P(A \text{ and } B) = P(A) \cdot P(B|A)$

• For independent events:

  $P(A \text{ and } B) = P(A) \cdot P(B)$

• Example: Probability that a randomly selected senior was matched with a residency position and it was one of their top three choices:

  • $P(A) = \frac{18,108}{19,326}$

  • $P(B|A) = 0.756$

  • $P(A \text{ and } B) = \frac{18,108}{19,326} \times 0.756 = 0.708$

• Complement: Probability that a matched senior did not get one of their top three choices:

  • $P(B'|A) = 1 - P(B|A) = 1 - 0.756 = 0.244$

Section 3.3 - The Addition Rule

Mutually Exclusive Events

• Two events are mutually exclusive if they cannot occur at the same time (no outcomes in common).

• Example: Rolling a 3 and rolling a 4 on a die are mutually exclusive events.

Addition Rule for Probability

• The probability that event A or B occurs:

  $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$

• For mutually exclusive events:

  $P(A \text{ or } B) = P(A) + P(B)$

• This rule can be extended to any number of mutually exclusive events.

Combining Rules and Complements

• Example: Probability that a draft pick is a running back (A) or wide receiver (B):

  • $P(A \text{ or } B) = P(A) + P(B) = \frac{16}{255} + \frac{37}{255} = \frac{53}{255}$

  • Probability that the draft pick is not a running back or wide receiver:

    $1 - P(A \text{ or } B) = 1 - \frac{53}{255} = \frac{202}{255} = 0.792$

Section 3.4 - Additional Topics in Probability and Counting

Permutations

• A permutation is an ordered arrangement of objects.

• The number of different permutations of n distinct objects is $n!$ (n factorial).

• Formulas:

  $n! = n \cdot (n-1) \cdot (n-2) \cdot (n-3) \cdots 1$

  $0! = 1$

• Examples:

  • $6! = 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 720$

  • $4! = 4 \cdot 3 \cdot 2 \cdot 1 = 24$

Permutations of n Objects Taken r at a Time

• The number of different permutations of n distinct objects taken r at a time:

• Formula:

  $nP_r = \frac{n!}{(n-r)!}$, where $r \leq n$

• Example: Number of four-digit codes with no repeated digits (n = 10, r = 4):

  $10P_4 = \frac{10!}{(10-4)!} = \frac{10!}{6!} = 10 \cdot 9 \cdot 8 \cdot 7 = 5040$ ways

Distinguishable Permutations

• Number of distinguishable permutations of n objects where $n_1$ are of one type, $n_2$ of another, etc.:

• Formula:

  $\frac{n!}{n_1! \cdot n_2! \cdot n_3! \cdots}$, where $n_1 + n_2 + n_3 + \cdots = n$

• Example: Arranging 6 one-story, 4 two-story, and 2 split-level houses (n = 12):

  $\frac{12!}{6! \cdot 4! \cdot 2!} = 13,860$ distinguishable ways

Combinations

• A combination is a selection of r objects from n objects, without regard to order.

• Formula:

  $nC_r = \frac{n!}{(n-r)! \cdot r!}$, where $r \leq n$

Additional info: These notes cover the foundational probability concepts, including classical, empirical, and subjective probability, as well as the basic counting principles, permutations, and combinations. These are essential for understanding more advanced topics in probability and statistics.