3.2 Independence of events

Independence of events is a crucial concept in probability theory. It occurs when one event doesn't affect the likelihood of another. This idea is key for calculating probabilities in many real-world situations, from coin flips to complex engineering systems.

Understanding event independence helps engineers solve problems involving multiple random occurrences. By knowing when events are independent, we can use simpler multiplication rules to find probabilities, making calculations easier and more accurate in various fields.

Independence of Events

Definition of event independence

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• Two events A and B are independent if the occurrence of one event does not influence the probability of the other event occurring
  • Expressed mathematically as $P(A|B) = P(A)$ and $P(B|A) = P(B)$, meaning the conditional probability of A given B is equal to the probability of A, and vice versa
  • Can also be written as $P(A \cap B) = P(A) \times P(B)$, indicating that the probability of the intersection of A and B is equal to the product of their individual probabilities (coin flips, rolling dice)
• For more than two events, independence requires that the occurrence of any subset of events does not affect the probability of the remaining events occurring
  • For instance, events A, B, and C are independent if $P(A \cap B \cap C) = P(A) \times P(B) \times P(C)$, meaning the probability of the intersection of all three events is equal to the product of their individual probabilities
  • Additionally, pairwise independence must hold for all pairs of events: $P(A \cap B) = P(A) \times P(B)$, $P(A \cap C) = P(A) \times P(C)$, and $P(B \cap C) = P(B) \times P(C)$ (drawing cards from a deck with replacement)

Identifying independent events

• To determine if events are independent, check whether the probability of the intersection of the events is equal to the product of their individual probabilities
  • If $P(A \cap B) = P(A) \times P(B)$, then events A and B are independent (selecting marbles from a bag with replacement)
  • If $P(A \cap B) \neq P(A) \times P(B)$, then events A and B are dependent, meaning the occurrence of one event affects the probability of the other event (selecting marbles from a bag without replacement)
• When dealing with more than two events, verify that the probability of the intersection of all events equals the product of their individual probabilities and that pairwise independence holds for all pairs of events
  • If these conditions are met, the events are independent (multiple spins of a roulette wheel)
  • If either condition is not satisfied, the events are dependent (drawing multiple cards from a deck without replacement)

Multiplication rule for probabilities

• The multiplication rule for independent events states that the probability of the intersection of independent events equals the product of their individual probabilities
  • For two independent events A and B, $P(A \cap B) = P(A) \times P(B)$ (probability of getting heads on two consecutive coin flips)
  • For n independent events $A_1, A_2, ..., A_n$, $P(A_1 \cap A_2 \cap ... \cap A_n) = P(A_1) \times P(A_2) \times ... \times P(A_n)$ (probability of rolling a specific sequence of numbers on multiple dice rolls)
• When calculating the probability of the union of independent events, combine the addition rule and the multiplication rule
  • For two independent events A and B, $P(A \cup B) = P(A) + P(B) - P(A \cap B) = P(A) + P(B) - P(A) \times P(B)$ (probability of getting at least one six in two rolls of a die)

Applications in engineering problems

• Identify the events in the problem and determine their independence
  • Look for keywords such as "not affected," "unrelated," or "regardless" to identify independent events (components failing independently in a system)
  • If the events are not explicitly stated as independent, use the definition of independence to check their relationship (probability of defects in manufactured items from different production lines)
• Apply the multiplication rule for independent events to calculate the required probabilities
  • If the problem asks for the probability of the intersection of independent events, multiply their individual probabilities (probability of multiple components functioning correctly in a system)
  • If the problem asks for the probability of the union of independent events, use the addition rule and the multiplication rule together (probability of at least one component failing in a redundant system)
• Interpret the results in the context of the engineering problem
  • Relate the calculated probabilities to the likelihood of the events occurring in the given scenario (assessing the reliability of a system based on the probabilities of component failures)
  • Use the results to make decisions or draw conclusions based on the problem's requirements (determining the number of redundant components needed to achieve a desired system reliability)