The addition rule for non-mutually exclusive events states that to find the probability of either event A or event B occurring, you must add the probabilities of both events and then subtract the probability of both events happening together. This rule is essential for accurately calculating probabilities when two events can occur at the same time, making it different from the addition rule for mutually exclusive events.
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The formula for the addition rule for non-mutually exclusive events is given by: P(A ∪ B) = P(A) + P(B) - P(A ∩ B).
The subtraction of P(A ∩ B) accounts for double counting when both events can happen simultaneously.
This rule is particularly important in situations where events share common outcomes, such as in card games or rolling dice.
When applying this rule, ensure that you have accurate data on both individual probabilities and their joint probability.
Understanding this rule helps clarify how different probabilities interact, especially in more complex probability scenarios.
Review Questions
How does the addition rule for non-mutually exclusive events differ from the rule for mutually exclusive events?
The main difference between the addition rule for non-mutually exclusive events and the one for mutually exclusive events lies in how probabilities are calculated. For mutually exclusive events, the formula is simply P(A ∪ B) = P(A) + P(B), as these events cannot occur together. However, in non-mutually exclusive cases, we must also subtract the joint probability P(A ∩ B) to avoid double counting outcomes where both events occur.
Why is it necessary to subtract the joint probability when using the addition rule for non-mutually exclusive events?
Subtracting the joint probability P(A ∩ B) is necessary because it corrects for over-counting scenarios where both events happen at the same time. If we simply added P(A) and P(B), we would count instances where both A and B occur twice. By subtracting this overlap, we ensure that our total probability accurately reflects the likelihood of either event occurring without duplication.
Evaluate a scenario where the addition rule for non-mutually exclusive events would be applicable and discuss its implications.
Consider a scenario involving a deck of cards where we want to find the probability of drawing either a heart or a queen. Both outcomes can occur simultaneously if we draw the queen of hearts. Using the addition rule, we calculate: P(heart) = 13/52, P(queen) = 4/52, and P(queen of hearts) = 1/52. Thus, applying the rule gives us: P(heart or queen) = (13/52 + 4/52 - 1/52) = 16/52. This example illustrates how recognizing overlapping outcomes allows us to determine accurate probabilities in combined event scenarios.
Related terms
Mutually Exclusive Events: Events that cannot occur at the same time, meaning the occurrence of one event excludes the possibility of the other event occurring.
Joint Probability: The probability of two events occurring simultaneously, which is crucial in calculating the addition rule for non-mutually exclusive events.
Probability: A measure of the likelihood that an event will occur, expressed as a number between 0 and 1.
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