3.3 Two Basic Rules of Probability

Probability rules are the building blocks of statistical analysis. They help us understand how likely events are to occur, whether alone or in combination with other events. These rules are crucial for making predictions and informed decisions in various fields.

The multiplication and addition rules for probabilities are key concepts. They allow us to calculate the likelihood of multiple events happening together or separately, considering whether events are independent, dependent, mutually exclusive, or non-mutually exclusive. Understanding these rules is essential for solving complex probability problems.

Probability Rules

Multiplication rule for event probabilities

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• Independent events
  • Events A and B are independent if the occurrence of one does not affect the probability of the other
  • Probability of both events occurring is the product of their individual probabilities: $P(A \text{ and } B) = P(A) \times P(B)$
    • Example: Probability of rolling a 6 on a fair die ($\frac{1}{6}$) and flipping a head on a fair coin ($\frac{1}{2}$) is $\frac{1}{6} \times \frac{1}{2} = \frac{1}{12}$
• Dependent events
  • Events A and B are dependent if the occurrence of one affects the probability of the other
  • Probability of both events occurring is the product of the probability of the first event and the conditional probability of the second event given the first: $P(A \text{ and } B) = P(A) \times [P(B|A)](https://www.fiveableKeyTerm:P(B|A))$
    • Example: Probability of drawing a heart from a standard deck of cards ($\frac{13}{52}$) and then drawing a king from the remaining cards ($\frac{3}{51}$) is $\frac{13}{52} \times \frac{3}{51} = \frac{1}{68}$
  • Conditional probability $P(B|A)$ is the probability of event B occurring given that event A has already occurred
    • Example: In the previous example, $P(\text{king}|\text{heart}) = \frac{3}{51}$ because there are 3 kings left out of 51 remaining cards after drawing a heart

Addition rule for event probabilities

• Mutually exclusive events
  • Events A and B are mutually exclusive if they cannot occur at the same time
  • Probability of either event occurring is the sum of their individual probabilities: $P(A \text{ or } B) = P(A) + P(B)$
    • Example: Probability of rolling an even number ($\frac{1}{2}$) or a prime number ($\frac{1}{2}$) on a fair die is $\frac{1}{2} + \frac{1}{2} = 1$
• Non-mutually exclusive events
  • Events A and B are non-mutually exclusive if they can occur at the same time
  • Probability of either event occurring is the sum of their individual probabilities minus the probability of both events occurring: $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$
    • Example: Probability of drawing a heart ($\frac{13}{52}$) or a face card ($\frac{12}{52}$) from a standard deck is $\frac{13}{52} + \frac{12}{52} - \frac{3}{52} = \frac{11}{26}$, where $\frac{3}{52}$ is subtracted to avoid double-counting the 3 cards that are both hearts and face cards (jack, queen, king of hearts)
  • This rule is closely related to the concept of complementary events, where the probability of an event not occurring is 1 minus the probability of it occurring: $P(\text{not A}) = 1 - P(A)$

Multiple event probability calculations

1. Identify whether the events are independent or dependent
   • If independent, use the multiplication rule for independent events
   • If dependent, use the multiplication rule for dependent events and calculate the necessary conditional probabilities
2. Identify whether the events are mutually exclusive or non-mutually exclusive
   • If mutually exclusive, use the addition rule for mutually exclusive events
   • If non-mutually exclusive, use the addition rule for non-mutually exclusive events and calculate the probability of the intersection of the events
3. Break down complex probability problems into simpler sub-problems
   • Identify the individual events and their probabilities
   • Determine the relationships between the events (independent, dependent, mutually exclusive, non-mutually exclusive)
   • Apply the appropriate probability rules and formulas based on the relationships between the events
   • Combine the results of the sub-problems to obtain the final probability
     • Example: Probability of drawing 2 aces from a standard deck without replacement can be broken down into (1) probability of drawing the first ace ($\frac{4}{52}$) and (2) probability of drawing the second ace given the first was already drawn ($\frac{3}{51}$), then multiplied together using the multiplication rule for dependent events to get $\frac{4}{52} \times \frac{3}{51} = \frac{1}{221}$
   • Tree diagrams can be useful for visualizing and calculating probabilities in multi-step problems

Probability Foundations and Visualization

• Sample space: The set of all possible outcomes in a probability experiment
• Set theory: The mathematical foundation for probability, dealing with collections of objects (events)
• Venn diagrams: Visual representations of sets and their relationships, useful for understanding probability concepts and solving problems involving multiple events
• The law of total probability: A fundamental rule that relates marginal probabilities to conditional probabilities, often used in conjunction with Bayes' theorem for more complex probability calculations