Mutually Exclusive Events

<a href="https://www.prepswift.com/quizzes/quiz/prepswift-mutually-exclusive-events" target="_blank">Mutually Exclusive Events Exercise</a>Mutually Exclusive Events are two events that CANNOT happen simultaneously. $A$ versus $not \ A$ The most common example of mutually exclusive events is "something" and "not something." For example, a ball cannot be "red" and "not red." It's either one or the other. In addition, because $A$ and $not \ A$ represent ALL possibilities, their sum equals $1$. $$P_A + P_{not \ A} = 1$$ The probability of $A$ happening and it not happening sums to $1$. For example, what is the probability of rolling an odd number or not an odd number in a fair, $6$-sided die? According to the above, their sum should be $1$. The odd numbers are $1$, $3$, and $5$ and the "not odd" numbers are $2$, $4$, and $6$. The odd numbers have a probability of $\frac{3}{6}$, or $0.5$. The "not odd" numbers have the same probability. Notice they sum to $1$. Mutually Exclusive versus Independent Events These concepts can sometimes be conflated. Notice the differences between the two. <ul> <li>Mutually Exclusive Events: "rolling a 4" and "rolling a 6" in one roll of the die.</li> <li>Independent Events: "rolling a 4" and then "rolling a 6" in two rolls of the die.</li> </ul> Here's another example: <ul> <li>Mutually Exclusive Events: "flipping heads" and "flipping tails" in one coin flip.</li> <li>Independent Events: "flipping heads" ten times in a row with ten coin flips.</li> </ul>