Mutually exclusive events, also known as disjoint events, are events that cannot happen at the same time. In other words, if one event occurs, the other cannot. These events are typically represented by the symbol “∩” in probability theory, indicating that they intersect or overlap.

One example of mutually exclusive events is flipping a coin. The two possible outcomes of flipping a coin are heads or tails. If the coin lands on heads, it cannot also land on tails, and vice versa. Therefore, the events of flipping a heads and flipping a tails are mutually exclusive.

Another example of mutually exclusive events is rolling a die. A die has six sides, each with a different number. If the die is rolled and lands on a 3, it cannot also land on a 4, 5, or 6. Therefore, the events of rolling a 3 and rolling a 4, 5, or 6 are mutually exclusive.

In both examples, the events are exclusive because only one outcome can occur at a time. In the case of flipping a coin, the coin cannot land on both heads and tails simultaneously. In the case of rolling a die, the die cannot land on multiple numbers at once.

Mutually exclusive events are important in probability because they help to determine the likelihood of an event occurring. For example, if a coin is flipped, the probability of flipping heads is 1/2 because there are two possible outcomes and one of them is heads. However, if a die is rolled, the probability of rolling a 3 is 1/6 because there are six possible outcomes and one of them is a 3.

In addition to calculating probabilities, mutually exclusive events are also used in decision making. For example, if a person is deciding whether to go to the beach or the mountains for their vacation, they cannot do both at the same time. Therefore, the events of going to the beach and going to the mountains are mutually exclusive. In this situation, the person must choose one event over the other.

Overall, mutually exclusive events are events that cannot happen simultaneously. They are important in probability and decision making because they help to determine the likelihood of an event occurring and guide choices between exclusive options.