13.7 Probability

Probability fundamentals form the backbone of statistical analysis. They help us quantify uncertainty and make predictions about future events. From basic models to complex calculations, these concepts are essential for understanding chance in everyday life and scientific research.

Probability rules and techniques provide tools for solving complex problems. By combining events, using complements, and applying counting methods, we can tackle a wide range of scenarios. These skills are crucial for making informed decisions in fields like finance, science, and engineering.

Probability Fundamentals

Basic probability model creation

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• Probability measures likelihood of an event occurring between 0 (impossible) and 1 (certain)
• Sample space (S) contains all possible outcomes of an experiment or random process (rolling a die)
• Event (E) is a subset of the sample space (rolling an even number)
• Probability of an event E denoted as $[P(E)](https://www.fiveableKeyTerm:P(E))$
• Calculate probability by dividing number of favorable outcomes by total number of possible outcomes $P(E) = \frac{\text{number of favorable outcomes}}{\text{total number of possible outcomes}}$ (probability of drawing a heart from a deck of cards is $\frac{13}{52} = \frac{1}{4}$)

Equal likelihood event probabilities

• When all outcomes in sample space equally likely, probability of event E is $P(E) = \frac{\text{number of outcomes in E}}{\text{total number of outcomes in S}}$
• Example: rolling a fair six-sided die, probability of rolling a 3 is $\frac{1}{6}$ (one favorable outcome out of six possible outcomes)
• Flipping a fair coin, probability of getting heads is $\frac{1}{2}$ (one favorable outcome out of two possible outcomes)

Probability Rules and Techniques

Union rules for combined events

• Union of events A and B, $A \cup B$, occurs when either A or B, or both, occur
• Probability of union of events A and B is $P(A \cup B) = P(A) + P(B) - P(A \cap B)$
  • $P(A \cap B)$ is probability of intersection of A and B (both occurring simultaneously)
• If events A and B mutually exclusive (cannot occur simultaneously), then $P(A \cap B) = 0$ and union probability simplifies to $P(A \cup B) = P(A) + P(B)$ (probability of drawing a heart or a spade from a deck of cards is $\frac{13}{52} + \frac{13}{52} = \frac{1}{2}$)
• Independence of events occurs when the occurrence of one event does not affect the probability of the other event

Complement rule in probability

• Complement of event A, $[A'](https://www.fiveableKeyTerm:A')$ or $[A^c](https://www.fiveableKeyTerm:A^c)$, occurs when A does not occur
• Probability of complement of event A is $[P(A')](https://www.fiveableKeyTerm:P(A')) = 1 - P(A)$
• Example: if probability of drawing a red card from a standard deck is $\frac{1}{2}$, then probability of not drawing a red card (complement) is $1 - \frac{1}{2} = \frac{1}{2}$
• Probability of not rolling a 6 on a fair six-sided die is $1 - \frac{1}{6} = \frac{5}{6}$

Counting techniques for complex probabilities

• Fundamental Counting Principle: if event A can occur in m ways and independent event B can occur in n ways, then two events can occur together in m × n ways (choosing a meal from a menu with 3 appetizers and 4 entrees results in 3 × 4 = 12 possible meal combinations)
• Permutations are arrangements of objects in a specific order
  1. Number of permutations of n distinct objects is $[n!](https://www.fiveableKeyTerm:n!) = n \times (n-1) \times (n-2) \times ... \times 3 \times 2 \times 1$ (number of ways to arrange 5 books on a shelf is $5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$)
  2. Number of permutations of n objects taken r at a time is $[P(n, r)](https://www.fiveableKeyTerm:P(n,_r)) = \frac{n!}{(n-r)!}$ (number of ways to select 3 people from a group of 10 to stand in a line is $P(10, 3) = \frac{10!}{(10-3)!} = \frac{10!}{7!} = 720$)
• Combinations are selections of objects without regard to order
  • Number of combinations of n objects taken r at a time is $[C(n, r)](https://www.fiveableKeyTerm:C(n,_r)) = \binom{n}{r} = \frac{n!}{r!(n-r)!}$ (number of ways to select 3 people from a group of 10 to serve on a committee is $C(10, 3) = \binom{10}{3} = \frac{10!}{3!(10-3)!} = \frac{10!}{3!7!} = 120$)
• Use counting techniques to calculate probabilities by determining number of favorable outcomes and total number of possible outcomes in complex scenarios (probability of drawing 2 aces from a deck of cards in 2 draws without replacement is $\frac{C(4, 2)}{C(52, 2)} = \frac{\binom{4}{2}}{\binom{52}{2}} = \frac{6}{1326} = \frac{1}{221}$)

Advanced Probability Concepts

• Conditional probability is the probability of an event occurring given that another event has already occurred
• Random variables are variables whose values depend on the outcome of a random experiment
• Expected value is the average outcome of an experiment if it is repeated many times
• The law of large numbers states that as the number of trials increases, the sample mean approaches the expected value
• Bayes' theorem is used to calculate conditional probabilities and update probabilities based on new information