An event in probability refers to a set of outcomes from a particular experiment or activity. For example, rolling a six-sided die has six possible outcomes. Each outcome (rolling a 1, 2, 3, 4, 5, or 6) is an event.

Inclusive Events

Inclusive events can occur simultaneously. They are not mutually exclusive and can happen at the same time within the same experiment.

Example: Consider drawing a card from a standard deck of 52 cards. The event of drawing a red card (26 outcomes) and the event of drawing a face card (Jack, Queen, King, 12 outcomes) are inclusive because you can draw a card that is both red and a face card (6 outcomes: the Jack, Queen, and King of Diamonds and Hearts).

Exclusive Events

Exclusive events, or mutually exclusive events, cannot happen at the same time. If one event occurs, the other cannot.

Example: Flipping a coin. The outcomes can either be heads or tails, not both. If you flip a coin once, getting heads excludes the possibility of getting tails from the same flip, making these events mutually exclusive.

Combining Events: Addition Rule

The addition rule helps calculate the probability of either of two events happening.

• For mutually exclusive events, the probability that Event A or Event B occurs is the sum of their individual probabilities: P(A∪B)=P(A)+P(B). For non-mutually exclusive (inclusive) events, you must subtract the probability of both events occurring together from the sum of their individual probabilities to avoid double-counting: P(A∪B)=P(A)+P(B)−P(A∩B). Examples to Illustrate

  1. Mutually Exclusive: The probability of rolling a 5 or a 6 on a six-sided die. Since these outcomes cannot occur simultaneously, P(5∪6)=P(5)+P(6)=61​+61​=31​.
     

  2. Inclusive: Consider a bag containing 20 marbles of various colors. If there are 8 red marbles and 5 blue marbles, and 3 marbles are both red and blue, the probability of drawing a red or blue marble is calculated by considering the overlap: P(Red∪Blue)=P(Red)+P(Blue)−P(Red∩Blue).
     

Conclusion

Understanding the distinction between inclusive and exclusive events, along with applying the addition rule appropriately, is crucial for accurate probability calculations. These concepts form the backbone of probabilistic reasoning, aiding in predicting outcomes and making informed decisions based on the likelihood of various events.