SOLUTION: P mutually exclusive and independent events - Studypool - Free Printable

Problem: Determine whether the following events are mutually exclusive. Answer "True" or "False" and provide a reason.

#### Definitions:
- Mutually Exclusive Events: Two events are mutually exclusive if they cannot occur at the same time. In other words, if one event happens, the other cannot happen.

---

Solution:

#### a) Die A lands on a six; die B lands on a four.
- Reasoning: These two events involve two different dice (Die A and Die B). The outcome of Die A does not affect the outcome of Die B. Both dice can show their respective numbers simultaneously.
- Answer: False
- Reason: The events can occur at the same time because they are independent of each other.

---

#### b) A randomly-selected playing card is a Heart; the same card is a red card.
- Reasoning: All Hearts in a standard deck of cards are red. Therefore, if a card is a Heart, it must also be red. These two events are not mutually exclusive because they can occur together.
- Answer: False
- Reason: A Heart is always a red card, so these events can occur simultaneously.

---

#### c) A coin lands Heads up; a coin lands Tails up.
- Reasoning: A single coin flip can result in either Heads or Tails, but not both at the same time. These two outcomes are mutually exclusive because they cannot occur simultaneously.
- Answer: True
- Reason: A coin cannot land on both Heads and Tails at the same time.

---

#### d) You have a cat; you have a goldfish.
- Reasoning: Having a cat and having a goldfish are two separate events that do not exclude each other. It is possible to own both a cat and a goldfish simultaneously.
- Answer: False
- Reason: Owning a cat does not prevent you from owning a goldfish, so these events can occur together.

---

#### e) The first ball you pick from a bag is green; the first ball you pick from a bag is red.
- Reasoning: When picking the first ball from a bag, it can only be one color at a time. If the first ball is green, it cannot be red, and vice versa. These two events cannot occur simultaneously.
- Answer: True
- Reason: The first ball picked can only be one color, so these events are mutually exclusive.

---

#### f) A die lands on an even number; a die lands on a prime number.
- Reasoning: The even numbers on a die are {2, 4, 6}, and the prime numbers on a die are {2, 3, 5}. The number 2 is both even and prime, so it is possible for a die to land on a number that is both even and prime. Therefore, these events are not mutually exclusive.
- Answer: False
- Reason: The number 2 is both even and prime, so these events can occur simultaneously.

---

Final Answers:

1. a) False
2. b) False
3. c) True
4. d) False
5. e) True
6. f) False

Boxed Final Answer:

\[
\boxed{
\begin{array}{ll}
\text{a) False} & \text{(Reason: Independent events)} \\
\text{b) False} & \text{(Reason: All Hearts are red)} \\
\text{c) True} & \text{(Reason: Coin cannot land on both Heads and Tails)} \\
\text{d) False} & \text{(Reason: Can own both a cat and a goldfish)} \\
\text{e) True} & \text{(Reason: Ball can only be one color)} \\
\text{f) False} & \text{(Reason: Number 2 is both even and prime)}
\end{array}
}
\]