Which of the Following Is the Best Statement of the Use of the Addition Rule of Probability?
The addition rule of probability is a fundamental concept in probability theory that allows us to calculate the probability of the occurrence of two or more events. It is a powerful tool used to determine the likelihood of multiple events happening simultaneously or sequentially. There are different ways to state the addition rule, but the best statement of its use can be summarized as follows:
“When two events A and B are mutually exclusive, the probability of either event A or event B occurring is equal to the sum of their individual probabilities.”
In other words, if two events cannot occur at the same time, then the probability of either event happening is the sum of their individual probabilities. This statement is concise, accurate, and captures the essence of the addition rule.
To illustrate this concept, let’s consider a simple example. Suppose we have a deck of cards, and we want to determine the probability of drawing either a red card or a face card. There are 26 red cards in the deck (13 hearts and 13 diamonds) and 12 face cards (4 jacks, 4 queens, and 4 kings). Since red cards and face cards are mutually exclusive (no card can be both red and a face card), we can use the addition rule to calculate the probability.
The probability of drawing a red card is 26/52 (since there are 26 red cards out of a total of 52 cards). Similarly, the probability of drawing a face card is 12/52. Using the addition rule, we can calculate the probability of drawing either a red card or a face card as follows:
P(red or face) = P(red) + P(face) = 26/52 + 12/52 = 38/52 = 0.73
Therefore, the probability of drawing either a red card or a face card is 0.73 or 73%.
Q: Can events A and B be dependent and still use the addition rule?
A: No, the addition rule is applicable only to mutually exclusive events. If events A and B are dependent, we need to use the multiplication rule to calculate their joint probability.
Q: What if events A and B are not mutually exclusive?
A: If events A and B can occur simultaneously, then we need to consider their intersection. In this case, the addition rule cannot be directly applied. We would need to use the inclusion-exclusion principle to calculate the probability of A or B.
Q: Can the addition rule be extended to more than two events?
A: Yes, the addition rule can be extended to any number of mutually exclusive events. The probability of any of the events occurring is equal to the sum of their individual probabilities.
Q: What if events A and B are not exclusive, but one event is a subset of the other?
A: In this case, we need to subtract the probability of the overlapping region to avoid double counting. This can be done using the subtraction rule of probability.
In conclusion, the best statement of the use of the addition rule of probability is that when two events A and B are mutually exclusive, the probability of either event A or event B occurring is equal to the sum of their individual probabilities. This rule is a powerful tool that allows us to calculate the likelihood of multiple events happening simultaneously or sequentially. By understanding and applying this rule correctly, we can make informed decisions and predictions based on probabilities.