What is Addition Rule For Probabilities?
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The addition rule for probabilities describes two formulas, one for the probability for either of two mutually exclusive events happening and the other for the probability of two non-mutually exclusive events happening.The first formula is just the sum of the probabilities of the two events. The second formula is the sum of the probabilities of the two events minus the probability that both will occur.
Definition
The Probability Addition Theorem is a fundamental theorem in probability theory used to calculate the probability of the occurrence of two events. It includes two formulas: one for mutually exclusive events and another for non-mutually exclusive events. For mutually exclusive events, the probability of either event occurring is the sum of their individual probabilities. For non-mutually exclusive events, the probability is the sum of their individual probabilities minus the probability of both events occurring simultaneously.
Origin
The origin of the Probability Addition Theorem dates back to the 17th century when probability theory began to form as a branch of mathematics. The correspondence between Blaise Pascal and Pierre de Fermat on gambling problems is considered the beginning of probability theory. Over time, the Probability Addition Theorem became a fundamental component of probability theory.
Categories and Features
The Probability Addition Theorem is mainly divided into two categories: mutually exclusive events and non-mutually exclusive events. Mutually exclusive events are those that cannot occur simultaneously, such as getting heads and tails in a coin toss. For mutually exclusive events, the theorem simply adds the probabilities of the two events. Non-mutually exclusive events are those that can occur simultaneously, such as rolling an even number and a number greater than three on a die. For non-mutually exclusive events, the theorem requires subtracting the probability of both events occurring to avoid double counting.
Case Studies
Case 1: Suppose there are 3 red balls and 2 blue balls in a bag, and one ball is drawn at random. Event A is drawing a red ball, and event B is drawing a blue ball. Since these events are mutually exclusive, using the Probability Addition Theorem, P(A or B) = P(A) + P(B) = 3/5 + 2/5 = 1. Case 2: Suppose in a deck of cards, event A is drawing a red card, and event B is drawing a king. Since these events are not mutually exclusive (as there are kings among the red cards), using the Probability Addition Theorem, P(A or B) = P(A) + P(B) - P(A and B) = 26/52 + 4/52 - 2/52 = 28/52.
Common Issues
Common issues investors face when applying the Probability Addition Theorem include confusing mutually exclusive and non-mutually exclusive events, which can lead to incorrect probability calculations. Another common misconception is ignoring the probability of simultaneous occurrence in non-mutually exclusive events, leading to double counting of probabilities.
