Mutually Exclusive Events Worksheet | Geometry PDF Worksheets - Free Printable

Problem Analysis and Solution

The worksheet focuses on understanding mutually exclusive and non-mutually exclusive events, as well as calculating probabilities. Let's break it down section by section.

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#### Section A: Mutually Exclusive vs. Non-Mutually Exclusive Events

We need to determine whether each pair of events is mutually exclusive or non-mutually exclusive.

1. Tossing a head with a coin and tossing a tails with a coin
- Mutually Exclusive: These events cannot happen at the same time. A coin can only land on heads or tails, not both.
- Non-Mutually Exclusive: Not applicable.

2. Throwing a six on a dice and throwing an even number on a dice
- Mutually Exclusive: No, because 6 is an even number. If you throw a 6, you have also thrown an even number.
- Non-Mutually Exclusive: Yes, because the event "throwing a six" is a subset of "throwing an even number."

3. Drawing a Spade from a pack of playing cards and drawing an Ace from a pack of playing cards
- Mutually Exclusive: No, because you can draw the Ace of Spades, which satisfies both conditions.
- Non-Mutually Exclusive: Yes, because the Ace of Spades is both a Spade and an Ace.

4. Drawing a red card from a pack of playing cards and drawing a Club from a pack of playing cards
- Mutually Exclusive: Yes, because Clubs are black cards, and red cards are Hearts and Diamonds. You cannot draw a card that is both red and a Club.
- Non-Mutually Exclusive: No.

5. Getting at least one head when two coins are thrown and getting at least one tails when two coins are thrown
- Mutually Exclusive: No, because you can get one head and one tail, which satisfies both conditions.
- Non-Mutually Exclusive: Yes, because the outcomes (HT, TH) satisfy both events.

6. Getting two prime numbers when two dice are thrown and getting a total score of four when two dice are thrown
- Mutually Exclusive: No, because the outcome (2, 2) is a prime number for both dice and sums to 4.
- Non-Mutually Exclusive: Yes, because (2, 2) satisfies both conditions.

7. Choosing two girls out of three girls and two boys and choosing one girl and one boy out of three girls and two boys
- Mutually Exclusive: Yes, because you cannot choose two girls and one girl and one boy simultaneously from the same group.
- Non-Mutually Exclusive: No.

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#### Section B: Completing the Table

We are given events A through F and need to determine whether pairs of events are mutually exclusive or non-mutually exclusive.

- Event A: The number chosen is odd.
- Event B: The number chosen is even.
- Event C: The number chosen is a multiple of 4.
- Event D: The number chosen is a factor of 24.
- Event E: The number chosen is a square number.
- Event F: The number chosen is a prime number.

##### Step-by-Step Analysis:

1. Event C and Event A
- Mutually Exclusive: Yes, because a multiple of 4 is always even, and an odd number cannot be a multiple of 4.
- Non-Mutually Exclusive: No.

2. Event D and Event C
- Mutually Exclusive: No, because some factors of 24 (e.g., 4, 8, 12, 24) are multiples of 4.
- Non-Mutually Exclusive: Yes.

3. Event F and Event B
- Mutually Exclusive: Yes, because a prime number (except 2) is always odd, and an even number cannot be prime (except 2).
- Non-Mutually Exclusive: No.

4. Event E and Event F
- Mutually Exclusive: Yes, because a square number greater than 1 is never prime (e.g., 4, 9, 16, etc.).
- Non-Mutually Exclusive: No.

5. Event F and Event D
- Mutually Exclusive: No, because some factors of 24 (e.g., 2, 3) are prime numbers.
- Non-Mutually Exclusive: Yes.

6. Event A and ?
- Mutually Exclusive: Yes.
- Non-Mutually Exclusive: Yes.
- Answer: Event B (The number chosen is even). Odd and even numbers are mutually exclusive and non-mutually exclusive in the context of the entire set of numbers.

7. Event C and ?
- Mutually Exclusive: Yes.
- Non-Mutually Exclusive: No.
- Answer: Event A (The number chosen is odd). A multiple of 4 is always even, so it cannot be odd.

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#### Extension: Probability Calculation

1. Calculate the probability of getting 2 or 3 on the spinner.
- The spinner has 8 equal sections.
- The numbers 2 and 3 appear once each.
- Total favorable outcomes: 2 (one 2 and one 3).
- Total possible outcomes: 8.
- Probability = \(\frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{2}{8} = \frac{1}{4}\).

2. Explain why the probability of getting a 2 or red is not \(\frac{3}{8} + \frac{3}{8}\).
- There are 3 red sections and 3 sections with the number 2.
- However, one of the red sections also has the number 2 (the section labeled "2" in red).
- If we simply add \(\frac{3}{8} + \frac{3}{8}\), we are double-counting the section that is both red and has the number 2.
- Correct calculation:
\[
P(\text{2 or red}) = P(\text{2}) + P(\text{red}) - P(\text{2 and red})
\]
\[
P(\text{2 or red}) = \frac{3}{8} + \frac{3}{8} - \frac{1}{8} = \frac{5}{8}
\]

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Final Answers

#### Section A
1. Tossing a head with a coin and tossing a tails with a coin: Mutually Exclusive
2. Throwing a six on a dice and throwing an even number on a dice: Non-Mutually Exclusive
3. Drawing a Spade and drawing an Ace: Non-Mutually Exclusive
4. Drawing a red card and drawing a Club: Mutually Exclusive
5. Getting at least one head and getting at least one tails: Non-Mutually Exclusive
6. Getting two prime numbers and getting a total score of four: Non-Mutually Exclusive
7. Choosing two girls and choosing one girl and one boy: Mutually Exclusive

#### Section B
1. Event C and Event A: Mutually Exclusive
2. Event D and Event C: Non-Mutually Exclusive
3. Event F and Event B: Mutually Exclusive
4. Event E and Event F: Mutually Exclusive
5. Event F and Event D: Non-Mutually Exclusive
6. Event A and Event B: Mutually Exclusive and Non-Mutually Exclusive
7. Event C and Event A: Mutually Exclusive

#### Extension
1. Probability of getting 2 or 3: \(\boxed{\frac{1}{4}}\)
2. Explanation for why \(P(\text{2 or red}) \neq \frac{3}{8} + \frac{3}{8}\): Double-counting the section that is both red and has the number 2. Correct probability is \(\frac{5}{8}\).