7.9 Conditional Probability and the Multiplication Rule

Conditional probability helps us calculate the likelihood of events happening when we already know something else has occurred. It's like figuring out the chances of rain when you see dark clouds. This concept is crucial in multistage experiments, where outcomes depend on previous events.

The multiplication rule for compound events lets us determine the probability of multiple things happening together. It's like calculating the odds of winning a game that requires multiple successful steps. This rule is especially useful when dealing with dependent events in real-world scenarios.

Conditional Probability and the Multiplication Rule

Conditional probabilities in multistage experiments

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• Conditional probability calculates the likelihood of an event occurring given that another event has already happened
  • Notation $[P(A|B)](https://www.fiveableKeyTerm:P(A|B))$ represents "the probability of A given B"
  • Formula $P(A|B) = \frac{P(A \cap B)}{P(B)}$ where $P(A \cap B)$ is the probability of both events A and B occurring together
• Multistage experiments consist of multiple steps or stages, each with its own set of possible outcomes
• Probability trees visually depict the possible outcomes and their probabilities in a multistage experiment
  • Branches represent possible outcomes
  • Probabilities are written along each branch
  • The probability of a specific path is the product of the probabilities along that path (coin toss, dice roll)
• Probability tables organize and calculate probabilities in multistage experiments
  • Rows and columns show possible outcomes at each stage
  • The probability of each outcome combination is the product of the probabilities in the corresponding row and column (weather forecast, medical diagnosis)
• Venn diagrams can be used to visualize the relationships between events and their probabilities in multistage experiments

Multiplication rule for compound events

• The multiplication rule calculates the probability of two events A and B both occurring as the product of the probability of A occurring and the conditional probability of B occurring given that A has occurred
  • Formula $P(A \cap B) = P(A) \times P(B|A)$
• For independent events, where the occurrence of one does not affect the probability of the other, the multiplication rule simplifies to $P(A \cap B) = P(A) \times P(B)$
• To determine the probability of compound events joined by "and":
  1. Identify the individual events and their probabilities
  2. Determine if the events are independent or dependent
  3. For independent events, multiply the individual probabilities
  4. For dependent events, calculate the conditional probability of the second event given the first, then multiply by the probability of the first event (drawing cards, rolling dice)
• The law of total probability can be used to calculate the probability of an event by considering all possible scenarios

Impact of new information on probabilities

• New information can alter the probability of an event occurring
• When additional information is provided, probabilities may need to be updated using conditional probability
  • Example: The probability of a randomly selected student being a senior is 0.25. If it is known that the selected student is female, and 60% of seniors are female, the updated probability of the student being a senior given that she is female can be calculated using conditional probability (medical test results, weather forecasts)
• Bayes' theorem updates probabilities based on new information
  • Formula $P(A|B) = \frac{P(B|A) \times P(A)}{P(B)}$
  • $P(A)$ is the prior probability of event A occurring before considering the new information
  • $P(B|A)$ is the likelihood of observing the new information (event B) given that event A has occurred
  • $P(B)$ is the probability of observing the new information (event B) regardless of whether event A has occurred (disease diagnosis, machine failure)

Foundations of Probability Theory

• Set theory provides the mathematical framework for understanding probability events and their relationships
• Probability axioms form the fundamental rules that govern probability calculations:
  1. The probability of any event is a non-negative real number between 0 and 1
  2. The probability of the entire sample space is 1
  3. For mutually exclusive events, the probability of their union is the sum of their individual probabilities