13.7 Probability

Probability concepts form the foundation for understanding random events and their likelihood. From basic models to advanced techniques, these tools help us quantify uncertainty in various scenarios, from simple coin flips to complex real-world situations.

Applying probability concepts involves identifying problem types, determining sample spaces, and using appropriate rules. Whether dealing with equally likely outcomes, unions, complements, or complex counting scenarios, these methods allow us to calculate and interpret probabilities in context.

Probability Concepts

Basic probability model creation

• Probability measures likelihood of an event occurring expressed as number between 0 (impossible event) and 1 (certain event)
• Probability models represent and analyze random phenomena (flipping a coin, rolling a die, drawing a card from a deck)
• Sample space ($S$) set of all possible outcomes of a random experiment
• Event ($E$) subset of the sample space
  • $P(E) = \frac{\text{number of favorable outcomes}}{\text{total number of possible outcomes}}$

Equal likelihood event probabilities

• When all outcomes in sample space are equally likely, probability of an event is ratio of number of favorable outcomes to total number of possible outcomes
  • Rolling a fair six-sided die, probability of rolling a 3 is $\frac{1}{6}$
• To calculate probability of an event with equally likely outcomes:
  1. Determine total number of possible outcomes (sample space)
  2. Count number of favorable outcomes (event)
  3. Divide number of favorable outcomes by total number of possible outcomes

Union rules for combined events

• Union of two events $A$ and $B$, denoted as $A \cup B$, is event that occurs when either $A$ or $B$, or both, occur
• Probability of union of two events $A$ and $B$ given by:
  • $P(A \cup B) = P(A) + P(B) - P(A \cap B)$
    • $P(A \cap B)$ probability of intersection of events $A$ and $B$, event that occurs when both $A$ and $B$ occur simultaneously
• For mutually exclusive events (events that cannot occur simultaneously), $P(A \cap B) = 0$, so union rule simplifies to:
  • $P(A \cup B) = P(A) + P(B)$

Complement rule in probability

• Complement of an event $A$, denoted as $A'$ or $A^c$, is event that occurs when $A$ does not occur
• Complement rule states sum of probabilities of an event and its complement is 1:
  • $P(A) + P(A') = 1$
• To find probability of an event using complement rule:
  1. Calculate probability of complement of the event
  2. Subtract complement's probability from 1

Counting techniques for complex probabilities

• Fundamental Counting Principle: If an event can occur in $m$ ways, and another independent event can occur in $n$ ways, then the two events can occur together in $m \times n$ ways
• Permutations: Number of ways to arrange $n$ distinct objects in a specific order
  • Number of permutations of $n$ objects taken $r$ at a time given by: $P(n, r) = \frac{n!}{(n-r)!}$
• Combinations: Number of ways to select $r$ objects from a set of $n$ distinct objects, where order of selection does not matter
  • Number of combinations of $n$ objects taken $r$ at a time given by: $C(n, r) = \binom{n}{r} = \frac{n!}{r!(n-r)!}$
• These counting techniques can be used to calculate probabilities in complex situations by determining number of favorable outcomes and total number of possible outcomes

Advanced Probability Concepts

• Conditional probability: Probability of an event occurring given that another event has already occurred
• Independence: Two events are independent if the occurrence of one does not affect the probability of the other
• Bayes' theorem: Formula used to calculate conditional probabilities
• Random variable: A variable whose possible values are outcomes of a random phenomenon
• Expected value: The average outcome of an experiment if it is repeated many times

Applying Probability Concepts

• Identify type of probability problem (equally likely outcomes, union of events, complement, or counting techniques)
• Determine sample space and number of favorable outcomes
• Apply appropriate rule or technique to calculate probability
  • For equally likely outcomes, use ratio of favorable outcomes to total outcomes
  • For union of events, use union rule: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$
  • For complement of an event, use complement rule: $P(A) + P(A') = 1$
  • For complex situations, use counting techniques (permutations or combinations) to determine number of favorable outcomes and total outcomes
• Interpret calculated probability in context of the problem