Answer:
To determine the probability of two events using the general addition rule, we need to know whether the events are mutually exclusive or not. Mutually exclusive events are events that cannot happen at the same time, such as rolling a 1 or a 6 on a single die. Non-mutually exclusive events are events that can happen at the same time, such as drawing a red card or an ace from a deck of cards.
The general addition rule for any two events A and B is:
P(A or B)=P(A)+P(B)−P(A and B)
where P(A and B) is the probability of both events happening together. This term is subtracted to avoid double-counting the outcomes that are in both A and B.
If the events are mutually exclusive, then P(A and B) is zero, because there is no overlap between the events. In this case, the general addition rule simplifies to:
P(A or B)=P(A)+P(B)
This means that the probability of either event happening is just the sum of their individual probabilities.
For example, suppose we roll a fair six-sided die once and want to find the probability of getting a 1 or a 6. These events are mutually exclusive, because we cannot get both numbers on one roll. Therefore, we can use the simplified addition rule:
P(1 or 6)=P(1)+P(6)
Since each number has an equal chance of 1/6, we can calculate:
P(1 or 6)=61+61=62=31
Therefore, the probability of getting a 1 or a 6 on one roll is 1/3.
If the events are not mutually exclusive, then P(A and B) is not zero, because there is some overlap between the events. In this case, we need to use the full general addition rule and subtract the probability of the overlap.
For example, suppose we draw one card from a standard deck of 52 cards and want to find the probability of getting a red card or an ace. These events are not mutually exclusive, because there are two cards that are both red and aces: the ace of hearts and the ace of diamonds. Therefore, we need to use the full general addition rule:
P(red card or ace)=P(red card)+P(ace)−P(red card and ace)
There are 26 red cards and 4 aces in the deck, so their probabilities are 26/52 and 4/52, respectively. There are 2 cards that are both red and aces, so their probability is 2/52. Therefore, we can calculate:
P(red card or ace)=5226+524−522=5228=137
Therefore, the probability of getting a red card or an ace on one draw is 7/13.
To summarize, the general addition rule can be used to find the probability of two events happening in either way: A or B. The rule depends on whether the events are mutually exclusive or not. If they are mutually exclusive, we simply add their probabilities. If they are not mutually exclusive, we add their probabilities and subtract their overlap.