A die is rolled, and the number that falls uppermost is observed. Let E denote the event that the number shown is even, and let F denote the event that the number is odd number.

A. Are the events E and F mutually excluseve?
B. Are the evnts # and F complementary?

Out of the possible numbers (in the sample space), if you cannot represent both events (odd and even) by a single outcome, then the events are mutually exclusive. In this case, we cannot have a number which is both odd and even, so E and F are mutually exclusive.

Events A and B are complementary if they are
1. mutually exclusive, AND
2. A∪B make up the sample space.

In the case of E and F, the sample space is
Ω={1,2,3,4,5,6},
E={2,4,6}
F={1,3,5}
so
E∪F={1,2,3,4,5,6}=Ω
Therefore E and F are complementary.

Thank you

You're welcome!

A. Are the events E and F mutually exclusive?

Well, it seems we have a bit of a situation here. E represents the event that the number shown is even, while F represents the event that the number is odd. Now, unless we're dealing with some crazy, funky, out-of-this-world die that can simultaneously land on both an even and an odd number, I'd say that E and F are indeed mutually exclusive. In other words, you either get an even number or an odd number, but never both at the same time. It's like trying to eat a pizza and a cupcake at the same time – you can't have your cake and eat it too, or in this case, have your even and odd numbers at the same time. So, E and F are definitely mutually exclusive!

B. Are the events E and F complementary?

Ah, the concept of complementarity. It's a bit like those friends you have who are complete opposites but somehow get along perfectly. Well, in this case, E and F are not complementary. Complementary events are when they are the only two possibilities and one event happening excludes the other from happening. For example, if we were rolling a dice and the only possible outcomes were either a 6 or not a 6, then those two events would be complementary. However, in our scenario, E and F are not the only two possibilities. We could also roll a 1, 3, or 5, which are neither even nor odd. So, sorry to burst your bubble, but E and F are not complementary. They're more like two funny clowns at a party – they might not be complementary, but they sure add some entertainment to the event!

A. In order to determine whether the events E and F are mutually exclusive, we need to understand their definitions.

The event E represents the occurrence of an even number when rolling a die, while the event F represents the occurrence of an odd number.

Since the numbers on a die can only be either even or odd, E and F are considered mutually exclusive events. This means that they cannot occur simultaneously. If an even number is rolled, it cannot be an odd number at the same time, and vice versa.

B. To determine whether the events E and F are complementary, we also need to understand the definition of complementary events.

Complementary events are two events that cover all possible outcomes and are mutually exclusive. In other words, when one event occurs, the other cannot occur.

In this case, if we define the event # as the occurrence of a number that is neither even nor odd (for example, a non-existent number), then E and # would be complementary events. If an even number is rolled, it cannot be a non-existent number, and vice versa.

However, the question asks about the events E and F. Since they are mutually exclusive, they are not complementary events. If an even number is rolled (event E), it is certain that the number is not odd (event F), and vice versa.